• To demonstrate an understanding of relations, functions, and function notation. Supported by learning objective 1.
• To recognize and graph algebraic transcendental and piecewise functions. Supported by learning objectives 2 and 3.
• To determine properties of functions from graphs and equations. Supported by learning objectives 4 and 6.
• To perform transformations and composition of functions and the graphs of functions. Supported by learning objectives 5 and 7.
• To support conclusions reached through effective and appropriate use of a graphing calculator. Supported by learning objectives 1 to 7.
| B.1 |
| B.2 |
| B.3 |
| B.4 |
| B.5 |
| B.6 |
| B.7 |
Objectives
B.1 Back to Top
To demonstrate a clear understanding of the terms function and relation, and to use function notation.
To discuss ideas using own language and then later incorporating the related mathematical terminology (COM).
To identify functions as being polynomial, constant, linear, quadratic, cubic, power, root, reciprocal, rational, algebraic, trigonometric, exponential, logarithmic, or transcendental, and to recognize the basic graph forms thereof.
To further develop knowledge, skills, and attitudes as lifelong learners (IL).
To demonstrate a consistent commitment to understanding their own emotions/feelings and their sources, and the abilities to use this understanding to support decision making, constructive social interactions, and strengthen learning (PSVS).
To engage in activities that require exploration in order to develop understandings of the concept of functions (CCT).
To represent understandings through a variety of communication modes (COM).
Instructional Notes
Functions and function notation play a dominant role throughout the development of Calculus understanding. Evaluating limits, determining continuity, finding derivatives, curve sketching, or solving optimization problems, all require the student to be proficient in communicating using functions and function notation.
Students first formally met functions and function notation in Math A30, although the terms linear function and quadratic function would have been used in Math 10 and Math 20. In Math B30, students were introduced to exponential, logarithmic, polynomial, and rational functions, as well as the reciprocal and the inverse of a function. Math C30 introduced students to the trigonometric functions.
It is important that students be able to describe the difference between a relation and a function. A relation is simply a set of ordered pairs while a function is a set of ordered pairs in which each x-coordinate is paired with one and only one y-coordinate .
The analogy of regarding a function as being a number changing machine is a good one. The numbers entering the machine, the input values , collectively form the domain , of the function. The numbers exiting the machine, the output values , also called function values , collectively form the range of the function. (Domain and range will be dealt with in detail in B.6.) The function converts one set (often x values) into another (often y values). As the value of y depends on the value of x , or is a function of the value of x , we write 
. Some common functions are the squaring function, the reciprocal function, the square root function, the sine function, and so on. The notation, 
, brings to mind the number x entering the machine, f through the hopper gate 
, and exiting as y . The notation 
, read “ f of 4 is 16”, can be viewed as 4 entering function f and exiting as 16. Perhaps f was the squaring function.
is the y value that is paired with x .
16 is sometimes called the image of 4. This brings to mind another analogy that can be used to describe a function, namely that of a lens that converts a slide's image to a screen image. In the case above, the lens (perhaps the squaring lens) converted the image on the slide, 4, into the screen image 16. By selecting a different lens, the slide's screen image can be changed.
The notation 
can also be thought of as “there is a function called f , that takes the number x and converts it into the number 
”. You might also read this as “the function f takes whatever and converts it into the square of whatever”. This helps the students to interpret 
as 
since the “whatever” in this instance is 
. Writing 
and thinking of w as “whatever” may also be useful.
Ask each student to think of a simple function (a number changing machine) without disclosing this function to anyone. Selecting a student, give him/her some input values and have the student give the corresponding output values—using the appropriate function vocabulary. (For example, tell the student that 
, and the student replies with “ f of 4 is 16”. See if anyone in the class can determine the function that the student has chosen. Repeat this exercise with other students (COM).
The symbol used to represent the input values of a function is known as the independent variable. The symbol used to represent the output values of a function is known as the dependent variable. If long distance calling charges are $0.10 per minute, then the cost of a long distance call is a function of the number of minutes that you talk. If C represents cost, and t represents the number of minutes, then 
. In this instance, t is the independent variable (it can change) and C is the dependent variable (its value depends on t .)
Students should be familiar with the following four ways of representing a function. For some functions, one form will be preferred over the others.
• words: “square the number you are given”
• table of values:
9
Clearly you could never give a complete table of values for the function described in words above.
• function notation: 
• graph:
Students should practise using function notation in the context of a variety of functions and values for x . Have students evaluate functions of variable expressions. Students should also be required to find an x value, given an 
value, both algebraically and graphically. Choose examples for x and 
that cause students to deal with quotients that are zero, undefined, and indeterminate.
Examples/Activities
1. What function keys are built into your calculator? (Examples: the square, square root, and tangent functions.)
2. Are all the keys on your calculator function keys?
3. Parking meters do not take every kind of coin. Similarly, some number changing machines do not take every kind of number. What numbers will the function 
accept? Explain.
4. Knowing that a function can only have one y value for each x value, sketch the graph of a relation that is not a function. What is the vertical line test and how does it justify your sketch?
5. Explain why the relation 
is not a function.
6. Explain why the relation 
is not a function.
7. Does the equation 
represent a function? Why or why not?
8. Suppose that only the hot tap of a shower is opened. The temperature of the water leaving the showerhead is a function of the time that the tap is left open. Sketch a graph of this function.
9. True or false?
(a) The notation 
means that the x value that is paired with a y value of 5 is 4.
(b) The notation 
means that the x value that is paired with a y value of 4 is 5.
(c) The notation 
means that the y value that is paired with an x value of 5 is 4.
(d) The notation 
means that the y value that is paired with an x value of 4 is 5.
10. Consider the function 
. If 
, then how many values of x can be found such that 
?
11. Write each of the following as functions.
(a) The area 
of a square in terms of its side length s .
(b) The circumference of a circle 
in terms of its radius r .
(c) The surface area of a cube 
in terms of its edge length e .
(d) The volume of a cube 
in terms of the area of one of its faces A .
(e) The area 
of an equilateral triangle in terms of its side length x .
(f) The area 
of a rectangle in terms of its width x , if the perimeter of the rectangle is 60 cm.
Assume that the range of each function is restricted to the real numbers.
1. If 
, find 
, 
, 
, 
.
2. If 
, find 
, 
, 
, 
.
3. If 
, find 
, 
, 
, 
.
4. If 
, find 
, 
, 
.
5. If 
, find 
, 
, 
.
6. If 
, find 
, 
, and 
.
7. If 
, find 
, 
, 
, 
, and 
.
8. If 
, find x if 
.
9. If 
, find t if 
.
10. If 
, find x if 
.
11. By studying the graph of 
over the domain 
below, find:
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) w , if 
and 
(h) d , if 
and 
(i) all values of k in the interval 
, for which 
.
12. True or false? The value of 
is the height of the graph above the x -axis at 
.
Suggestions/Extensions
Ask the students to justify in as many ways as they can why a relation such as x 2 + y 2 = 100 is not a function (CCT). This can be done visually, abstractly, algebraically logically, etc. Continued experiences relating all methods of representation and analysis will strengthen the students' understanding of function and relations.
Encourage the students to create and share and solve problems that highlight real-life situations that involve relations that are not functions as well as ones that are functions.
Have the students research how to represent functions using an arrow (mapping) diagram (IL).
Suppose a function 
satisfies the conditions that 
, and 
for all x . Find:
(a) 
(b) 
(c) 
(d) 
.
Given that 
, find two values of c that satisfy 
= 
.
Have the students become familiar with how functions are composed to make a different function by identifying and defining functions within a function.
Objectives
B.2 Back to Top
To identify functions as being polynomial, constant, linear, quadratic, cubic, power, root, reciprocal, rational, algebraic, trigonometric, exponential, logarithmic, or transcendental, and to recognize the basic graph forms thereof.
To make and justify decisions based upon understanding of calculus concepts (CCT).
To further develop knowledge, skills, and attitudes as lifelong learners (IL).
To demonstrate a consistent commitment to understanding their own emotions/feelings and their sources, and the abilities to use this understanding to support decision making, constructive social interactions, and strengthen learning (PSVS).
To engage in activities that require exploration in order to develop understandings of the concept of functions (CCT).
To represent understandings through a variety of communication modes (COM).
Instructional Notes
For effective communication and understanding throughout the course, students need to be able to identify types of functions by examining the equation form of a function. It is also important that students know the graph forms of the different types of functions and be able to identify what type of function may be represented, given the graph of the function. The use of graphing calculators is highly recommended to review old ideas and establish the new ideas of this section.
A polynomial function is a function of the form 
where n is a nonnegative integer and the real numbers 
, 
, 
, . . . , 
, 
, 
are constants called the coefficients of the polynomial. The polynomial function is said to have degree n and leading coefficient 
. Thus 
is a polynomial function of degree 7 and leading coefficient 
. Polynomial functions of degree 0 are called constant functions and can be written in the form 
. Polynomial functions of degree 1 can be written in the form 
and are called linear functions (slope is m , y -intercept is b ). Polynomial functions of degree 2 can be written in the form 
and are called quadratic functions . Graphs of quadratic functions are called parabolas . Polynomial functions of degree 3 are called cubic functions .
Remind the students (Math B30) that polynomial functions of:
odd degree with 
begin in quadrant III and end in quadrant I.
odd degree with 
begin in quadrant II and end in quadrant IV.
even degree with 
begin in quadrant II and end in quadrant I.
even degree with 
, begin in quadrant III and end in quadrant IV.
A power function is a function that can be written in the form 
, where a is a real number. Students should be familiar with the following three cases.
Case One : 
, where n is a positive integer.
If a is a positive integer, the power function is also a polynomial function. Students should recognize that the shape of 
depends on whether n is even or odd. Emphasize that if n is even and n increases, the shape of 
is like that of 
except that the graphs will become increasingly flatter for x near 
and increasingly steeper for 
. Likewise if n is odd and n increases, then the shape of 
is like that of 
except that the graphs will become increasingly flatter for x near 
and increasingly steeper for 
. Students should recognize that all power functions pass through the points 
and 
. Students must be able to sketch the graphs of 
, 
, and 
effortlessly.
Case Two : 
, where n is a positive integer greater than 1.
The function 
is known as a root function . If 
, 
, the square root function with 
. If 
, 
, the cube root function with 
. By experimenting with the graphing calculator, have students discover that if n is even and increases, the shape of 
is like that of 
except that the graphs become steeper near 
and flatter for 
. If n is odd and increases, the shape of 
is like that of 
except that the graphs become steeper near 
and flatter for 
. Students must be able to sketch the graphs of 
and 
effortlessly.
.
The function 
is the reciprocal function whose graph is a rectangular hyperbola with the coordinate axes as asymptotes (Math C30).
A rational function is a function formed by taking the ratio of two polynomials. From Math B30, students should recall that 
has vertical asymptote lines at values of x for which 
, provided 
. If the degree of 
is less than the degree of 
, 
has horizontal asymptote 
, and if the degree of 
is equal to the degree of 
, then 
has horizontal asymptote 
, where k is the ratio of the leading coefficients of the polynomial functions. Note that the reciprocal function is also a rational function.
Algebraic functions are functions that are formed by performing a finite number of algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) with polynomials. Thus all rational functions are also algebraic. Some examples of algebraic functions are: 
, 
, and 
. The graphs of algebraic functions vary widely.
The functions 
, 
, 
, 
, 
, and 
are trigonometric functions . Students should have no problem graphing two class periods of each, unless they are taking Math C30 concurrently with Calculus. If so, do not take Calculus class time to develop what will happen in C30.
Exponential functions are functions of the form 
where the base, b , is a positive real number other than 1. Graphs of exponential functions of this form always pass through the point 
and lie entirely in quadrants II and I. If 
, the graph is always increasing, and if 
, the graph is always decreasing. The x-axis is a horizontal asymptote line.
Logarithmic functions are functions of the form 
, where the base, b , is a positive number other than 1. Graphs of logarithmic functions of this form always pass through the point 
and lie entirely in quadrants I and IV. If 
, the graph is always increasing, and if 
, the graph is always decreasing. The y-axis is a vertical asymptote line.
Transcendental functions are functions that are not algebraic. They include the trigonometric functions, the exponential functions, and the logarithmic functions.
is a polynomial function, a power function, and a rational function ( 
). Hence the intersection of the three sets. 
is a rational function, a polynomial function, 
, but not a power function. 
is a power function but not a rational nor a polynomial function.
Students could be divided into groups to research each of the different types of functions and then make a brief presentation to the class (IL, PSVS, CCT, COM). Students should demonstrate behaviours and attitudes required for working with and for others (CD3.2).
Examples/Activities
Students will need exercises to recognize the different kinds of functions.
Give an example (if possible) of:
1. a polynomial function of degree 3.
2. a linear function with negative slope.
3. a constant function whose graph passes through quadrants III and I.
4. a constant function whose graph passes through quadrants III and IV.
5. a quadratic function that opens downwards.
6. an increasing exponential function.
7. a decreasing exponential function.
8. an increasing logarithmic function.
9. a decreasing logarithmic function.
10. a power function that is a polynomial function.
11. a power function that is not a polynomial function.
Indicate whether the given statement is always true, sometimes true, or never true (CCT).
1. Linear functions are constant functions.
2. Quadratic functions are of degree 3.
3. The graph of a polynomial function of degree 4 with a negative leading coefficient will start in quadrant III and finish in quadrant IV.
4. The graphs of rational functions have vertical asymptote lines.
5. The graphs of rational functions have horizontal asymptote lines.
6. Trigonometric functions are periodic—their graphs repeat themselves.
7. The graph of a trigonometric function will contain a vertical asymptote line.
8. The graph of a trigonometric function will contain a horizontal asymptote line.
9. The reciprocal function is a power function.
Classify each of the following functions as being constant, identity, linear, quadratic, cubic, polynomial (state the degree), power, root, reciprocal, rational, algebraic, trigonometric, logarithmic, exponential, or transcendental. Use the most specific classification possible.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(j) 
(k) 
(l) 
(m) 
(n) 
(o) 
(p) 
(q) 
(r) 
Students should be able to recognize the different kinds of functions by examining the graphs of the functions.
Without the use of a graphing calculator, match each function with one of the two graphs immediately below. The window shown is 
for both x and y .
1. 2.
(a) 
(a) 
(b) 
(b) 
3. 4.
(a) 
(a) 
(b) 
(b) 
Draw the graphs of the following functions using a suitable calculator window. Describe how the graphs are positioned relative to one another for 
and for 
.
(a) 
, 
, 
(b) 
, 
, 
(c) 
, 
, 
(Recall the change of base formula:
.)
(d) 
, 
, 
Students should have instant recall of the graphs of the following basic functions in order to draw the graphs of similar functions by performing geometric transformations – objective B.5. Students could make “flash cards” showing the function on one side and graph on the other side. Later, in section B.4, students can add to their flash cards by classifying the functions as being increasing, decreasing, even, odd, one-to-one, or many-to one, and after section B.6, the domain and range could be added.
( k is a constant)
(later in the course, 
)
(later in the course, 
)
(from section A.5)
Suggestions/Extensions
Ask the students to justify the rules they have about where a polynomial function starts and ends on the basis of its leading coefficient and degree. It is important for students to become comfortable with thinking about large values – both positive and negative.
Have the students collect/research real-life data and then create a scatter plot representing the data. Have the students discuss what function or set of functions would best describe the data. The students could then use a graphing calculator to find the regression equation for a function. As more of a challenge, provide the students with data that would best be described by a piecewise function.
Ask the students to predict and then verify on their calculators what would need to be done to an exponential function in order for it to pass through the point (2, 0). Have the students share their reasoning (COM).
Have students research other functions such as:
• What other functions, found on your calculator but not in this section, are transcendental?
• What is the hyperbolic sine function?
• What is a supply function?
• What is a demand function? (IL)
Objectives
B.3 Back to Top
To determine function values for, and draw the graphs of, piecewise and step functions.
Instructional Notes
Students have had little or no experience with piecewise and step functions. These functions must be examined because they describe many phenomena and provide a helpful context from which to study continuity and differentiability.
A piecewise function is one that uses different function rules for different parts of the domain. The function 
describes the cost (in dollars) of renting a car and travelling x kilometres. The function indicates that if one rents the car and travels between 0 and 100 kilometres inclusive, the cost will be $40. One is charged, however, $0.25 for every kilometre beyond 100. Thus 
, while 
. To graph such a function, draw the graph of each piece of the function using the accompanying function rule. Use open and closed dots to indicate whether the endpoints of each piece are included or excluded. Be careful to never have two closed dots one above the other. The graph of this function, as drawn with a TI-83 calculator (in DOT MODE), appears below. Note how the function has to be entered into the calculator.
The graph of the function 
is shown below.
Encourage students to plot the point determined by the endpoint of each interval and use an open or closed circle to indicate whether the endpoint is included. Students may wish to use a table of values to find points.
To find function values, students must identify the interval in which the particular x value lies, and then use the appropriate formula. Thus for the function 
above, 
, because 
; 
because 
; 
because 
. Note: f(1) = 1 2 not 1-1 because of the domain. Similarly f(-4) = -4 - 1 not -2 because of the domain.
Note that the absolute value function is a piecewise function.
.
It is important for the students to be comfortable with how to read piecewise functions and to construct and read their graphs. It is also important for the students to understand and describe real-world problems that are represented by piecewise functions.
Examples/Activities
An excellent example of a piecewise function is that of the federal income tax system wherein the amount paid in tax is a function of one's income. Each income bracket pays tax at a different rate. You could draw a graph of this function (look up the tax rates for the current year on the Internet) and draw the graph of this function. A discussion of income tax would likely be an automatic outcome of this activity. Students could examine how different work scenarios can affect life scenarios (CD3.9)
If
,
evaluate each of the following:
(a) 
(b) 
(c) 
(d) 
(e) 
Draw the graph of the function 
above.
In the fall of 2003 the cost, 
, of mailing a standard size letter in Canada of mass m grams is given by the following postage function :
.
Explain why this is a piecewise function. Draw the graph of this function.
If your long distance calling charges are 10¢ per minute or portion thereof, write a piecewise function to describe this charge and draw the graph of the function. Use 
to represent the cost, in cents, of a phone call that lasts x minutes.
The signum function is defined as
.
This function is used to indicate whether a number is positive, negative, or zero. Sketch the graph of the function.
Suppose your annual dollar salary, 
, x months from today is given by the function 
(the exponent contains the greatest integer function). What does this mean is happening to your salary? Is this realistic? Use this function to determine your salary in:
(a) 0 mo. (b) 11 mo. (c) 13 mo. (d) 23 mo. (e) 255 mo.
Draw the graph of 
. How many years of working does this represent? What real-life conditions regularly have this type of graph associated with them?
Write a piecewise function to describe each graph. Each square denotes one unit.
1. 2.
Graph the following.
1. 
2. 
3. 
Write the equation for a real-world situation that involves a piecewise function. Explain what each piece of the function represents.
Suggestions/Extensions
Draw the graph of the function in the example at left using your graphing calculator. You will need to research how to enter piecewise functions. If you have a TI-83 search the web for “piecewise functions on the TI-83”.
Have the students create and identify real-life examples of piecewise functions.
Objectives
B.4 Back to Top
To identify functions as being even, odd, increasing, decreasing, one-to-one, and many-to-one, and the associated graph implications.
To listen, read, and view ideas and concepts analytically and critically (CCT).
Instructional Notes
Knowledge of whether a function is even, odd, increasing, decreasing, one-to-one, or many-to-one assists in understanding the graph of a function or determining if the inverse of a function is also a function. The graphing calculator will help in the exploration of this concept.
An understanding of symmetry is essential. Begin by showing students a design (portion of a graph) drawn in the first quadrant and ask them to reflect the design about the y -axis. Ask the students where the mirror image of every point 
in the first quadrant is in the second quadrant 
. Ask the students where the mirror image of every point 
in the first quadrant is in the third quadrant 
.
In discussing y -axis symmetry, stress that the y value at 
is the same as the y value at a . In discussing origin symmetry, stress that the y value at 
, is the opposite of the y value at a . This should then lead to the following definitions. A function is said to be an even function of x if 
for all values of x that lie in the domain of the function, thus an even function always has y -axis symmetry. A function is said to be an odd function of x if 
for all values of x that lie in the domain of the function, and thus an odd function always has origin symmetry. If you have knowledge of the graph of an even or odd function in the first quadrant, the graph can be completed by an appropriate reflection. The names even and odd can be related to the fact that power functions like 
and 
(note the even powers) are even functions because 
, namely 
and 
. Power functions like 
and 
(note the odd powers) are odd functions because 
, namely 
and 
. You may want to have the students draw the graphs of these functions on their graphing calculators to see the different symmetries.
Students should be able to determine if a function is even or odd by examining the graph of the function or by examining the function itself. To determine if the function 
is even or odd, you must examine 
. In this case 
which is the same as 
. Thus the function is even and its graph will have y -axis symmetry.
The function 
is odd and has origin symmetry because 
. The function 
is neither even nor odd because 
is not the same as 
nor is it equal to 
.
Examine the graphs of the trig functions and determine which are even and which are odd. Next, discuss whether each function is even or odd by examining the function itself. The identities 
and 
, taken in Math C30, will aid in the discussion.
Ask the students why a function cannot be symmetric about the x -axis.
A function is increasing on an interval I if 
whenever 
in the interval I . A function is decreasing on an interval I , if 
whenever 
in the interval I .
is decreasing 
is increasing
As can be seen below, the function 
is decreasing on the interval 
and increasing on the interval 
. The function 
is increasing on the interval 
.
Show the students the graphs of several functions. For the ones that are one-to-one say “these are”, for the others say “these are not”. Ask the students for the common characteristic of each group (CCT). This should lead to the definition that a function is one-to-one if it never has the same y value twice. In other words, 
if 
for all x values in the domain of the function. Discuss the graphical implications – a horizontal line will cross a one-to-one function only once. A function that is not one-to-one is a many-to-one function . Discuss the graphical implications. Clearly 
is many-to-one while 
is one-to-one.
Examples/Activities
Ask the students to use the graphs of the basic functions studied in B.3 to determine if the function is even, odd, one-to-one, or many-to-one. Have them add this information to their flashcards.
Show the students graphs of functions, other than the basic ones above, and have students determine if the function is even, odd, increasing, decreasing, one-to-one, or many-to-one. Students could create functions on their calculators and working with a partner, classify their own and their partner's functions.
A function has a domain 
and a portion of the graph is shown below. If the function is defined for all real values of x , complete the graph of the function if it is (a) even, (b) odd, (c) neither.
Find the coordinates of each point Q that is symmetric with point P if Q and P are symmetric about (a) the y -axis; (b) the origin.
1 . 
2 . 
3 . 
4 . 
5 . 
Classify each function as even, odd, or neither without drawing the graph of the function.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
For each graph, ask the students to use the graphs of the basic functions studied in B.3. For each graph, have students determine the intervals in which the function is increasing and the intervals in which the function is decreasing. Have students add this information to their flashcards.
Give the interval(s) over which each function is increasing or decreasing by examining the graph.
1. 2.
Divide the class into groups and assign each group one of the following statements. Ask them to explain why they think the statement is true (CCT, PSVS). Students should demonstrate helping skills such as problem solving, guidelines and consensus building (CD3.2).
1. A function that is increasing or decreasing for all real numbers could be an odd function.
2. A function that is increasing or decreasing for all real numbers could be an even function.
3. Even functions could never be one-to-one.
4. Odd functions could be many-to-one.
5. All functions are either even or odd.
6. An even polynomial function will have terms of only even degree.
7. An odd polynomial function cannot have a constant term.
Suggestions/Extensions
Have the students consider whether a function can be both even and odd.
Ask the students if the graph of an odd function passes through the origin.
(Careful, if the domain is appropriately restricted, the answer is no. Consider the function whose graph appears below. It has origin symmetry. Thus, it is odd.)
If an odd function passed through 
, then it would also have to pass through 
creating two points one above the other. You would no longer have a function. The graph below is not that of an odd function, because it is not even the graph of a function. For 
, there are two y values.
Ask the students if they can create other odd or even relations that are not functions. Have the students consider whether it is possible for a function to have x -axis symmetry. Challenge the students to write the equation of a relation with x -axis symmetry.
Have the students research what a strictly monotonic function is?
Ask students to sketch a graph that is one-to-many and ask them to explain/define what they see.
Objectives
B.5 Back to Top
To recognize the new function form and draw the graph of 
after it has been transformed by a shift, stretch, compression, or reflection about an axis.
Instructional Notes
Students need to recognize and graph functions that are simple transformations of the functions introduced in B.2.
Shifting, stretching, compressing, or reflecting about an axis creates a new function from an old one. This section investigates how these transformations change the original function and how a function must be changed to bring about a given transformation. Students must have access to a graphing calculator or graphing software in order to develop this section efficiently. Begin by having the students draw the graph of one of the basic graph forms. You might want to use 
or 
, as these are less familiar to students and do not exhibit y -axis symmetry. If you reflect the graph of an even function about the y -axis, the reflection will coincide with the original function and you will not observe any transformation. If students have already taken Math C30, then using 
as your basic function could also work well.
Vertical Shifts: Ask the students to graph 
using the standard window. Follow this by having students graph functions of the form 
and 
where 
. That is, have students graph functions like 
, 
, 
, 
. They should now have five graphs on their calculators. From their observations, students should conclude that if 
, they can obtain the graph of:
by shifting the graph of 
c units upwards.
by shifting the graph of 
c units downwards.
Horizontal Shifts: Ask the students to graph 
using the standard window. Follow this by having students graph functions of the form 
and 
where 
. That is, have students graph functions like 
, 
, 
, 
. From their observations students should conclude that if 
, they can obtain the graph of:
by shifting the graph of 
c units to the left.
by shifting the graph of 
c units to the right.
Vertical Stretches: Starting with 
, have students graph functions of the form 
where 
. That is, have students graph functions like 
, 
, 
. From their observations students should conclude that if 
, they can obtain the graph of 
by stretching vertically the graph of 
by a factor of c . That is, all the y values are c times higher than before.
Vertical Compressions: Starting with 
, have students graph functions of the form 
where 
. That is, have students graph functions like 
, 
, 
. From their observations, students should conclude that if 
, they can obtain the graph of 
by compressing vertically the graph of 
by a factor of c . That is, all the y values are 
times as high as they were before.
Horizontal Compressions: Starting with 
, have students graph functions of the form 
where 
. That is, have students graph functions like 
, 
, 
. From their observations, students should conclude that if 
, they can obtain the graph of 
by compressing horizontally the graph of 
by a factor of c . That is, the function reaches its former y values c times sooner. For example, the graph of 
has a y value of 2 if 
whereas 
has a y value of 2 if 
.
Horizontal Stretches: Starting with 
, have students graph functions of the form 
where 
. That is, have students graph functions like 
, 
, 
. From their observations, students should conclude that if 
, they can obtain the graph of 
by stretching horizontally the graph of 
by a factor of c . That is, the function reaches its former y values c times later. For example, the graph of 
has a y value of 2 if 
whereas 
has a y value of 2 if 
.
With stretches and compressions, the general effect should be emphasized rather than focusing on exact value changes.
Reflections About the x -Axis: Starting with functions such as 
, 
, 
have students graph 
, 
, 
. From their observations, students should conclude that they can obtain the graph of 
by reflecting the graph of 
about the x -axis.
Reflections About the y -Axis: Starting with functions such as 
, 
, 
, have students use a graphing calculator to graph 
, 
, 
. From their observations, students should conclude that the graph of 
can be obtained by reflecting the graph of 
about the y -axis.
With student input, create the following two pictorial summaries by giving students the graph of 
over a closed interval and asking them to position the graphs of:
(i) 
, 
, 
, and 
.
(ii) 
, 
, 
, 
After students understand single transformations, introduce multiple transformations. For example, if the graph of 
is given, ask students how it should be transformed to obtain the graph of 
, 
, 
,
Examples/Activities
A function 
passes through the points A 
, B 
, C 
, and D 
. Find the new coordinates of these four points if subjected to each of the following transformations.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(j) 
If 
find each of the following functions.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(j) 
If the function 
is transformed as described, what will be the equation of the new function created?
(a) 5 units upward.
(b) 5 units downward.
(c) 5 units left.
(d) 5 units right.
(e) stretched horizontally by a factor of 5.
(f) compressed horizontally by a factor of 5.
(g) stretched vertically by a factor of 5.
(h) compressed vertically by a factor of 5.
(i) reflected about the x -axis.
(j) reflected about the y -axis.
How would you transform the graph of 
to obtain the graph of each of the following functions?
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(j) 
Beginning with the graph of 
, have students draw the graphs of each of the following.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(j) 
Shown below is the graph of 
. Sketch the graph of each of the following functions.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
Beginning with the graph of 
, how would you transform it to obtain the graphs of the following?
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
Shown in the top left corner below is the graph of 
. Use transformation and determine the equation of the graph of each of the other functions.
Suggestions/Extensions
You can use the TI-83 calculator to see what happens under a given transformation as follows. Suppose that 
and you would like to see the graph of 
. Enter 
, and for 
, enter 
. Note that you find the symbols 
and 
by pressing VARS and selecting Y-VARS. The resulting graphs are shown below.
It is beneficial for students to relate the impact of a transformation by considering what impact it has on the point where x =0 in the original function. Questions like what value of x will behave like f( 0 ) in the original function help the students to understand rather than just memorize the different transformations. It also engages the students in a deeper understanding of the functions and variables in general.
Have the students relate their prior knowledge of linear, quadratic, and trignometric functions and the rules for graphing those functions to transformations of the function.
Working in pairs, have the students each sketch two graphs – one of a function f(x) and another function g(x) that is the result of one or more transformations on f(x) . The students exchange with their partners and describe the transformation shown by using f(x) and function notation.
Objectives
B.6 Back to Top
To find the domain and range of a function from the graph of the function and from the equation for the function.
To represent problems and understandings through a variety of communication modes (COM).
To make and justify decisions based upon understanding of calculus concepts (CCT).
Instructional Notes
Students generally have little trouble finding the domain and range of a function if they are able to examine the graph of the function. Finding the domain and range by just examining the function provides them with a much greater challenge. They will need to use the skills developed in each section of unit A, together with estimation skills and common sense.
Graphically: Have the students clarify the terms domain and range that were introduced in B.1. If you are looking at the graph of a function, you can find the domain by examining just how far left and how far right the graph extends, as the domain consists of the x values that can be placed in the number changing machine (function). You can find the range by examining just how high up and how low down the graph extends, since the range consists of the values that result after coming out of the number changing machine – the y values. Students could be divided into groups, and each group could be given the graphs of two or three functions. These graphs should include step functions and piecewise functions. Ask the groups to determine the domain and range of each function, writing their answers using set and interval notation. Students should later report their findings to the class. Also ask the students to draw the graphs of functions having a given domain and range. Again, answers can be shared with the class. Clearly answers will vary (COM, CCT).
From The Function: When students can determine domain and range by examining the graph of a function, move to determining the domain and range by just examining the function. This becomes difficult for students because they have worked with so many functions for which the domain has been all real numbers, that they have not become critical thinkers in this area.
It might be wise to work with the domain first – it is easier. Starting with linear, quadratic, and cubic functions, ask the students if there are any input values for which it would be impossible to determine an output value. Of course, there are none. It should be easy for students to come to the conclusion that the domain of every polynomial function is the set of real numbers.
Next examine rational functions such as 
, 
, 
. Again, ask the same question, “Are there any input values, for which we cannot determine the output?” Hopefully, students will recognize that they must not allow any value of x that causes the denominator to be 0. Students will have to use their factoring skills.
Now introduce some root functions such as 
, 
, 
, 
, 
, 
, 
, 
. Students should realize that it is impossible to find the even root of a negative number. Thus, they must solve a series of inequalities such as 
, 
, 
, and so on. The solutions to these inequalities form the domain.
Next examine root functions with odd indices such as 
, 
, 
. As the odd root index of any number can be determined, the domain of these three functions will be all real numbers.
Continue by examining exponential functions such as 
and 
. Once again, we can always find an output value for any input given, so these two functions have all real numbers as their domain.
Follow this by examining some logarithmic functions. Students will have to recall from Math B30 (also B.2) that they can only determine logarithms of positive numbers. Thus to find the domain of functions such as 
or 
, students will have to solve the accompanying inequalities 
and 
.
As it is possible to determine the sine and cosine of any real number, functions such as 
or 
have all real numbers as their domain. To determine the domain of trig functions such as 
, it is easiest to rewrite them in terms of the sine and cosine functions. Thus 
. Now we must avoid division by 0, so we need to know the values of 
for which 
. Since 
for multiples of 
, we can find the restrictions on x by solving 
where k is any integer. Thus, the domain is 
.
Finally, consider functions such as 
. It is possible to raise 2 to any real value, but the domain must exclude 
as this value would cause the exponent, 
, to be undefined. Consider 
. Although it is possible to find the cube root of any number, we can find the logarithm of only positive numbers. Thus 
, so 
.
Domain Summary
• You cannot divide by zero.
• You cannot take the even root of a negative number.
• You cannot find the logarithm of a non-positive number.
Finding The Range
There is no rule for finding the range of a function. Generally students need to be asking themselves questions such as:
• What happens to the value of the function for large positive x values?
• What happens to the value of the function for large negative x values?
• What happens to the value of the function near to any values in the domain that cause the denominator of the function to be zero?
• Do the numerator, denominator, or any part of the expression ever reach a minimum/maximum value?
• Determining the horizontal and vertical asymptote lines (Math B30) together with a sign analysis is helpful for rational functions.
Examples/Activities
Have the students determine the domain and range of the basic functions drawn on their flash cards in B.2 and add this information to the flash cards.
Find the domain and range of each function shown.
(a) (b)
(c) (d)
(e) (f)
Determine the domain and range of each of the following functions.
1. 
2. 
3. 
(complete 4. 
the square)
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 
Sketch the graph of a function having the given domain and range.
1. 
, 
2. 
, 
3. 
, 
Suggestions/Extensions
The following imagery may help students better understand domain and range. Suppose beams of light parallel to the y -axis shine down and up towards the x -axis. If the graph of the function blocks off the light's path, then that part of the x -axis that is in shade is the domain of the function. Similarly, if beams of light parallel to the x -axis shine right and left towards the y -axis, the range is that part of the y -axis that is in the shade.
Students should realize that the domain of odd powered polynomial functions will always be all real numbers while the range of even powered polynomial functions will be of the form:
• 
if the leading coefficient is positive ( m is the minimum value of the function)
• 
if the leading coefficient is negative ( M is the maximum value of the function).
Determining m and M will be decidedly easier when derivatives are understood.
Given the circle with equation 
, find the equation of the bottom half of the circle.
The parabola 
opens to the right. Find the equation of the top half of parabola.
Find the range of the function 
. You may want to rewrite the function as a piecewise function containing no absolute value signs.
Students should be encouraged to pull together all their prior knowledge about expressions and functions when determining domain and range. Students who wish to sketch what they know of the graph of a given function are exhibiting a powerful ability to combine their prior knowledge in order to solve problems.
Objectives
B.7 Back to Top
To build functions from other functions through function addition, subtraction, multiplication, division, and composition.
To use language as a tool for learning and communicating (COM).
Instructional Notes
Combinations of Functions
Students may be aware (from Math A30) that functions can be combined using the operations of addition, subtraction, multiplication, and division in much the same way we combine real numbers. Students may not be familiar with how to determine the domain of combined functions from the domains of the original functions. Illustrating with a specific example is most helpful for students.
Let f and g be two functions with domains A and B respectively. Then we define the functions 
, 
, 
, and 
as follows:
. The domain of 
is 
.
. The domain of 
is 
.
. The domain of 
is 
.
. The domain of 
is 
.
Thus if 
with domain 
and 
with domain 
, then:
with domain 
.
with domain 
.
with domain 
.
with domain 
. Note that 
has been excluded from 
because if 
, 
which is undefined.
If you know the graphs of 
and 
you can obtain the graph of 
by graphical addition (adding the corresponding y-coordinate s).
Function Composition
A thorough understanding of function composition greatly assists in using the chain rule (differentiation) and performing u substitution (integration).
Students have touched upon function composition in Math A30.
The composition of functions 
and 
, denoted by the symbol 
and read “ f of g ”, is the new function that is created by replacing x in function f by 
. That is 
. The output values from function g , become the input values for function f as diagrammed below.
Function Composition
x enters the g function machine and is turned out as 
. Then 
enters into the f function machine and is turned out as 
. This process is represented by 
.
The following definitions are extremely critical.
The domain of 
is the set of all values in the domain of g such that 
is in the domain of f .
The domain of 
is the set of all values in the domain of f such that 
is in the domain of g .
Similarly we define 
as 
. x enters the f machine and is changed into 
. 
enters the g machine and is changed into 
.
Using a pair of functions such as 
and 
, ask the students to find 
. Stress that since 
, students should first find 
, and having obtained 
, then evaluate 
. Be sure students are communicating clearly. The format could be 
. Next ask the students to compute 
. Note that the result, 
, is not the same as 
. Thus function composition is not commutative . Some more numerical examples should follow such as 
and 
. Students should also try 
and 
to realize that the domain of 
is definitely not all real numbers.
Next, determine 
. The function f takes anything and changes it into the square root of 4 less than the anything. Thus 
. Underlining (or shading) 
is helpful – students can more readily see 
as a “sort of” x . As the domain of 
is all real numbers, we can find the domain of 
by finding the domain of 
which is 
.
Similarly 
. One would be tempted, by looking at the result, 
, to assert that the domain of 
is all real numbers, but such is not the case even though 
is defined for all real numbers. By definition, the domain of 
is the set of all values in the domain of f such that 
is in the domain of g . Since f 's domain is 
, then we must choose those values in 
for which 
will be in the domain of g . Since g can accept any value, the domain of 
is 
.
Students should be able to recognize a given function as the composition of other functions. For example, if 
, then 
can be written as 
where 
and 
. Students should be encouraged to ask themselves, “What happened to x ?” In this case, one might say that first we have taken five more than three times x . Secondly, the square root of that result was taken. Clearly this could be viewed as the composition of three functions. A student could say that the first x was tripled; secondly, that the result was increased by 5; finally, the square root of the second result was taken. Thus 
where 
, 
, and 
.
Examples/Activities
1. If 
and 
, find each of the following.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
2. On copies of the grid below, sketch the graph of 
, 
, 
, 
, 
, 
.
3. Using the basic graphs of 
and 
, sketch the graph of 
.
Students should be given several questions like the one that follows:
1. If 
and 
, find each of the following.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
(j) 
2. Repeat question 1 if 
and 
.
3. Repeat question 1 if 
and 
.
4. Repeat question 1 if 
and 
.
The following exercise is known as function decomposition .
Find functions 
and 
so that 
. There may be more than one solution. Do not use the trivial solution 
and 
.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
Find functions 
, 
, and 
so that 
. There may be more than one solution. Do not use the trivial solution 
, 
and 
.
(a) 
(b) 
(c) 
(d) 
(e) 
If 
and 
, find a formula and the domain and range for 
and 
.
If 
and 
, find a formula and the domain and range for 
and 
.
If 
and 
, find a formula and the domain and range for 
and 
.
Suggestions/Extensions
You can graph combinations of functions and composition of functions using a graphing calculator. If 
and 
, then to graph 
using a TI-83 calculator with the standard window, enter the functions as shown.
Find the domain, range, and formulas for 
if 
and 
.
A graphic artist is drawing a square using computer software. The length of the side of the square in cm is 
, where t is the number of seconds after the artist begins drawing the square. The area of the square is 
. Find and interpret 
.
A crew is paving a rectangular patio and the length in metres of the paved rectangle as a function of time is given by 
. The width (in metres) of the rectangle as a function of time is given by 
. Find 
and interpret the result.
The number of teachers needed to staff a school of x students is given by the function 
. The number of vice-principals needed for a school having t teachers is given by the function 
. Find 
and interpret the result.
Have students create and identify a variety of examples that involve the composition of functions. Provide the students with a function such as 
and ask them to define two functions such that their composition gives f(x) .
If 
, find a function 
so that 
. (Hint: ask yourself what has to be done to function g to obtain function h ?)
1. 
2. 
3. 
4. 
Ask the students to try to find a pair of functions that are commutative with respect to composition. That is, find f(x) and g(x) such that 
. Have students discuss how the functions are related (COM).
See Appendix A for additional objective and activities.