Section 6.1 Page 310 to 322

The Exponential Function and Its Inverse

6.1

Investigate 1 What is the nature of the rate of change of an exponential function?

A: Numerical Analysis: Average Rate of Change

1. a) Copy and complete the table of values for the function y � 2x. Leave room for three more columns.

b) Describe any patterns you see in the values of y as x increases.

Tools• computer with The Geometer’s

Sketchpad®• graphing calculator (optional)

• grid paper

Exponential functions are useful for describing relationships. If the growth of a population is proportional to the size of the population as it grows, we describe the growth as exponential. Bacterial growth and compound interest are examples of exponential growth. Exponential decay occurs in nuclear reactions and in the depreciation in value of vehicles or equipment.

How can the inverse of an exponential function be found, and why is it useful? Graphing technology is useful for exploring the nature of the inverse of an exponential function. Applications of the inverse of an exponential function will appear throughout this chapter.

x y

0 1

1 2

2

3

4

5

6

310 MHR • Advanced Functions • Chapter 6

2. a) Calculate values for Δ1y, the fi rst differences, and record them in the third column.

b) Explain how these values confirm your answer to step 1b).

3. a) Compare the pattern of values for y and Δ1y. Explain what you notice.

b) Predict the pattern of values for the second differences, Δ2y, of this function.

c) Calculate the second differences and record them in the fourth column of the table. Was your prediction correct? Explain.

4. Predict the values for the third differences, Δ3y, and justify your prediction with mathematical reasoning. Test your prediction and enter these values in the last column.

5. Repeat steps 1 to 4 for the function y � 3x.

6. Ref lec t What do these results illustrate about the rate of change of an exponential function?

B: Graphical Analysis: Comparing Average and Instantaneous Rates of Change

1. Open The Geometer’s Sketchpad® and begin a new sketch.

2. a) Plot the function y � 2x.b) Construct two points on

the graph and label them A and B.

c) Construct a secant through A and B and measure its slope.

x y �1y �

2 y �

3 y

0 1

1 2

2

3

4

5

6

Technology Tip s

You can determine fi rst diff erences

with a graphing calculator:

Enter the values for x and y into

L1 and L2 of the List Editor.

Place the cursor on L3 in the

table and press O o

for [LIST].

From the OPS menu, choose

7:�List( and type O 2 for

[L2] and then a closing bracket.

Press e.

Technology Tip s

For instructions on how to plot a function

and perform other basic functions using

The Geometer’s Sketchpad®, refer to the

Technology Appendix on page 505.

6.1 The Exponential Function and Its Inverse • MHR 311

3. a) Click and drag the points so that A is at x � 0 and B is at x � 1.

b) What is the average slope of the curve for the interval [0, 1], correct to one decimal place?

c) How is the average slope related to the first of the first differences you found in part A of this Investigate?

4. a) Drag point B until it is very close to A, fi rst to the left, and then to the right, so that the line through A and B approximates a tangent to the curve at x � 0. Note the slope in each case and estimate the instantaneous rate of change when x � 0, correct to one decimal place.

b) Move both points very close to where x � 1 and estimate the instantaneous rate of change when x � 1, correct to one decimal place.

5. Repeat steps 3 and 4 for several points on the graph. Use whole-number values of x. Record all your results in a table like this one.

6. Ref lec t Compare the average rates of change to the fi rst differences you found in part A of this Investigate. What do you notice? Explain this relationship.

7. a) How are consecutive values of the instantaneous rate of change, mA and mB, related to each other in each case?

Interval Average Rate of

Change, mAB

Instantaneous Rate

of Change at A, mA

Instantaneous Rate

of Change at B, mBA B

x � 0 x � 1

x � 1 x � 2

x � 2 x � 3

x � 3 x � 4

x � 4 x � 5


b) Use linear interpolation (averaging) to estimate the instantaneous rate

of change at x � 1 _ 2 . Do you think this value is correct? Explain why

or why not.

c) Drag A and B close to where x � 1 _ 2 and check your estimate. Was your

estimate correct? If not, explain why not.

8. Ref lec t Explain how the results of step 7 illustrate that an exponential function has an instantaneous rate of change that is proportional to the function itself (i.e., also exponential).

9. Do these results hold true for different bases?

a) Explore this question for exponential functions, y � bx, i) with other values of b � 1 ii) with values of b, where 0 � b � 1

b) Reflec t Write a summary of your findings.

Example 1 Write an Equation to Fit Data

Write an equation to fi t the data in the table of values.

Solution

Calculate Δ1y to determine if the data represent an exponential function.

Because y is increasing at a rate proportional to the function, the function is exponential.

Consider the equation y � bx. Substitute the given values into this equation to fi nd b.

1 � b0 This statement is true for any value of b. 4 � b1

41 � b1

4 � b The only valid value for b is 4.

Check the other values in the table to make sure 4 is the correct value for b.

16 � b2

42 � b2

64 � b3

43 � b3

An equation for a function that fi ts the data in the table is y � 4x.

x y

0 1

1 4

2 16

3 64

x y �1y

0 1

1 4 3

2 16 12

3 64 48

C O N N E C T I O N S

The fi rst diff erences are

proportional to successive

y-values because their ratios

are equal. i.e.

4

_ 16

� 3

_ 12

and 16

_ 64

� 12

_ 48


1. Begin a new sketch with The Geometer’s Sketchpad®.

2. a) Plot the function y � 2x.

b) Identify the key features of the graph.

i) domain and range

ii) x-intercept, if it exists

iii) y-intercept, if it exists

iv) intervals for which the function is positive and intervals for which it is negative

v) intervals for which the function is increasing and intervals for which it is decreasing

vi) equation of the asymptote

3. a) Use the key features to sketch a graph of the function y � 2x in your notebook.

b) Draw the line y � x on the same graph. Explain how you can use this line to sketch a graph of the inverse, x � 2y.

c) Sketch a graph of the inverse function.

d) Verify that your sketch is accurate by comparing points on the two graphs. The x-coordinates and y-coordinates should be switched.

4. Use The Geometer’s Sketchpad® to verify your sketch by tracing the inverse of y � 2x as follows.

• Select the graph of f(x). From the Construct menu, choose Point On Function Plot.

• Select the constructed point. From the Measure menu, choose Abcissa (x). Repeat to measure Ordinate (y).

• Select the Ordinate (y) measure followed by the Abcissa (x) measure. From the Graph menu, choose Plot as (x, y).

• Select the image point that appears. From the Display menu, choose Trace Plotted Point.

• Click and drag the original constructed point on y � 2x along the graph of f(x) until a smooth curve is traced out.

5. a) Describe the shape of the pattern of points that appears.

b) Identify the key features of the inverse graph.

i) domain and range






Investigate 2 What is the nature of the inverse of an exponential function?

Tools• computer with The Geometer’s

Sketchpad®• grid paper

C O N N E C T I O N S

The abscissa and ordinate

are the x-coordinate and

y-coordinate, respectively,

of an ordered pair. For example,

in (1, �5), the abscissa is 1

and the ordinate is �5.


6. a) Copy and complete the table of values.

b) Verify that each ordered pair in the fourth column of the table is on your graph of the inverse function.

7. Ref lec t

a) How are the two graphs related?

b) When the image point was constructed from the constructed point on the graph of y � 2x, why were the coordinates chosen in the order (y, x)?

8. a) Using graphing technology, explore the effect of changing the base of the function y � bx for values of b for

i) b � 1

ii) 0 � b � 1

b) Reflec t Describe what happens to the graphs of the function and its inverse in each case.

xCalculation

2x � yy

Inverse

(y, x)

�4 2�4 � 1 _ 16

1 _ 16

( 1 _ 16

, �4)

�3 2�3 � ?

�2

�1

0

1

2

3

4


Example 2 Graphing an Inverse Function

Consider the function f(x) � 4x.

a) Identify the key features of the function.

i) domain and range






b) Sketch a graph of the function.

c) On the same set of axes, sketch a graph of the inverse of the function.

d) Identify the key features, as in part a) i) to vi), of the inverse of the function.

Solution

a) i) domain {x ∈ �}; range {y ∈ �, y > 0}

ii) no x-intercepts

iii) y-intercept 1

iv) positive for all values of x

v) increasing for all intervals

vi) horizontal asymptote with equation y � 0; no vertical asymptote

b), c)

d) i) domain {x ∈ �, x � 0}; range {y ∈ �}

ii) x-intercept 1

iii) no y-intercepts

iv) positive for x � 1 and negative for x � 1

v) increasing for all intervals

vi) vertical asymptote with equation x � 0; no horizontal asymptote

y

x2 4�2�4

4

2

�2

�4

0

y � 4x

x � 4y


<< >>KEY CONCEPTS

An exponential function of the form y � bx, b � 0, b � 1, has

• a repeating pattern of fi nite differences

• a rate of change that is increasing proportional to the function for b � 1

• a rate of change that is decreasing proportional to the function for 0 � b � 1

An exponential function of the form y � bx, b � 0, b � 1,

• has domain {x ∈ �}

• has range {y ∈ �, y � 0}

• has y-intercept 1

• has horizontal asymptote at y � 0

• is increasing on its • is decreasing on its domaindomain when b � 1 when 0 � b � 1

The inverse of y � bx is a function that can be written as x � by. This function

• has domain {x ∈ �, x � 0}

• has range {y ∈ �}

• has x-intercept 1

• has vertical asymptote at x � 0

• is a refl ection of y � bx about the line y � x

• is increasing on its domain • is decreasing on its domainwhen b � 1 when 0 � b � 1

y

x

y � bx, b � 1

0

y

x

y � bx, 0 � b � 1

0

y

x

y � bx, b � 1

x � by0

y

x

y � bx, 0 � b � 1

x � by

0


Communicate Your Understanding

C1 Explain how you can recognize whether or not a function is exponential by examining its

a) fi nite differences

b) graph

C2 Is the inverse of y � 2x a function? Explain your answer using

a) algebraic reasoning

b) graphical reasoning

C3 Consider the function y � 2x and its inverse. Describe the ways in which they are

a) alike

b) different

C4 a) What happens to the shape of the graph of f(x) � bx when b � 1?

b) What happens to the shape of its inverse?

c) Explain why this happens.

A Practise

For help with questions 1 and 2, refer to Investigate 1, part A.

1. Which of the following functions are exponential? Explain how you can tell.

A

B

C

D

2. Refer to question 1. For the exponential functions that you identifi ed, write an equation to fi t the data.

For help with questions 3 and 4, refer to Investigate 1, part B.

3. a) Use Technology Graph the function y � 1.5x over the domain 0 � x � 6 using graphing technology.

b) Determine the average rate of change of y with respect to x for each interval.

i) x � 1 to x � 2 ii) x � 2 to x � 3 iii) x � 3 to x � 4 iv) x � 4 to x � 5

x y

1 3

2 6

3 9

4 12

x y

0 0

1 1

2 3

3 10

4 16

x y

0 1

1 3

2 9

3 27

4 81

x y

�3 27

�2 9

�1 3

0 1

1 1 _ 3


c) Estimate the instantaneous rate of change of y with respect to x at each of the endpoints in part b).

d) Describe how these rates are changing over the given domain.

4. Repeat question 3 for the function y � ( 1 _ 2 )

x

over the domain �4 � x � 2.

For part b) use the following intervals. i) x � �3 to x � �2 ii) x � �2 to x � �1 iii) x � �1 to x � 0 iv) x � 0 to x � 1

For help with questions 5 to 8, refer to Investigate 2.

5. a) Copy the graph.

b) Write an equation for this exponential function.

c) Graph the line y � x on the same grid.

d) Sketch a graph of the inverse of the function by refl ecting its graph in the line y � x.


b) Write an equation for this function.

c) Graph the line y � x on the same grid.

d) Sketch a graph of the inverse of the function by refl ecting its graph in the line y � x.

7. Match each equation to its corresponding graph.

i) y � 5x ii) y � ( 1 _ 2 )

x

iii) y � 2x iv) y � ( 1 _ 5 )

x

a)

b)

c)

d)

y

x2 4�2�4

4

2

�2

�4

0

y

x2 4�2�4

4

6

8

2

0

y

x2 4�2�4

4

6

8

2

0

y

x2 4�2�4

4

2

�2

�4

0

y

x2 4�2�4

4

2

�2

�4

0

y

x2 4�2�4

4

2

�2

�4

0


8. Tell which graph from question 7 each graph below is the inverse of.

a)

b)

c)

d)

y

x2 4 6 8

4

2

�2

�4

0

y

x2 4 6 8

4

2

�2

�4

0

y

x2 4 6 8

4

2

�2

�4

0

y

x2 4 6 8

4

2

�2

�4

0

B Connect and Apply

9. Consider the functions f(x) � 3x, g(x) � x3, and h(x) � 3x.

a) Graph each function.

b) Make a list of the key features for each function, as in Investigate 2, step 2 b). Organize the information in a table.

c) Identify key features that are common to each function.

d) Identify key features that are different for each function.

e) How do the instantaneous rates of change compare for these three functions?

10. An infl uenza virus is spreading through a school according to the function N � 10(2)t, where N is the number of people infected and t is the time, in days.

a) How many people have the virus at each time?

i) initially, when t � 0 ii) after 1 day

iii) after 2 days iv) after 3 days

b) Graph the function. Does it appear to be exponential? Explain your answer.

c) Determine the average rate of change between day 1 and day 2.

d) Estimate the instantaneous rate of change after

i) 1 day ii) 2 days

e) Explain why the answers to parts c) and d) are different.

Use the functions f(x) � 4x and g(x) � ( 1 _ 2 )

x

to answer questions 11 to 18.

11. a) Sketch a graph of f.

b) Graph the line y � x on the same grid.

c) Sketch the inverse of f on the same grid by refl ecting f in the line y � x.

12. Identify the key features of f.

a) domain and range

b) x-intercept, if it exists

c) y-intercept, if it exists


d) intervals for which the function is positive and intervals for which it is negative

e) intervals for which the function is increasing and intervals for which it is decreasing

f) equation of the asymptote

13. Repeat question 12 for the inverse of f.

14. a) Sketch a graph of the function g.


c) Sketch the inverse of g on the same grid by refl ecting g in the line y � x.

15. Identify the key features of g.

a) domain and range

b) x-intercept, if it exists

c) y-intercept, if it exists

d) intervals for which the function is positive and intervals for which it is negative

e) intervals for which the function is increasing and intervals for which it is decreasing

f) equation of the asymptote

16. Repeat question 15 for the inverse of g.

17. Compare the graphs of f and g. Describe how they are

a) alike

b) different

18. Repeat question 17 for f�1 and g�1.



c) Graph the inverse of this function by refl ecting it in the line y � x.

d) Write an equation for the inverse function.

20. Write an equation for the inverse of the function shown.

21. Consider the equation f(x) � (�2)x.

a) Copy and complete the table of values.

b) Graph the ordered pairs.

c) Do the points form a smooth curve? Explain.

d) Use technology to try to evaluate

i) f (�0.5) ii) f (�2.5)

Use numerical reasoning to explain why these values are undefi ned.

e) Use these results to explain why exponential functions are defi ned only for functions with positive bases.

22. Chapter Problem A spaceship approaches Planet X and the planet’s force of gravity starts to pull the ship in. To prevent a crash, the crew must engage the thrusters when the ship is exactly 100 km from the planet. The distance away from the planet can be modelled by the function d(t) � (1.4)t, where d represents the distance, in hundreds of kilometres, between the ship and the planet and t represents the time, in seconds.

a) What is the ship’s average velocity between 1 s and 2 s? between 3 s and 4 s?

b) What is the ship’s instantaneous velocity at 3 s? at 4 s?

Connecting

Problem Solving

Reasoning and Proving

Reflecting

Selecting ToolsRepresenting

Communicating

x � 6y

y

x2 4 6 8

4

2

�2

�4

0

y

x2 4 6 8 10

4

2

�2

�4

0

x � � �y

110

x y

0

1

2

3

4


C Extend and Challenge

23. a) For the graph of any function f(x) � bx and its inverse, describe the points where the x-coordinates and y-coordinates are equal. Explain how the functions relate to the line y � x at these points.

b) Does your answer to part a) differ when b � 1 versus when 0 � b � 1?

24. Use Technology Open The Geometer’s Sketchpad® and begin a new sketch.

a) From the Graph menu, choose New Parameter. Call the parameter b and set its initial value to 2.

b) Plot the function f(x) � bx. Explore the shape of this graph and its inverse, using different values of b.

c) For which values of b is f

i) a function?

ii) undefi ned?

d) For which values of b is the inverse of f

i) a function?

ii) undefi ned?

e) Are the answers to parts c) and d) the same? Explain.

C A R E E R C O N N E C T I O N

Andre completed a 4-year bachelor of science degree in nuclear medicine at the Michener Institute for Applied Health Sciences. He now works in a hospital as a nuclear medicine technologist. After administering a dose of a radiopharmaceutical to a patient, he monitors the spread of the radioactive drug with a gamma scintillation camera. Andre saves the images on a computer. They will later be interpreted by a doctor. It is critical that Andre decide on the correct radioactive material to use, as well as calculating and preparing the proper dosage.

<<P6-8: photo of Andre, African male

with a white lab coat on>>

Technology Tip s

To graph a function using a parameter, choose Plot New Function from

the Graph menu, and double-click on the parameter b to change its

value when the Edit Parameter Value box appears.

The parameter b can be adjusted manually either by right-clicking on it

and choosing Edit Parameter or by selecting it and pressing � or �.

The parameter b can be adjusted dynamically by right-clicking on it

and choosing the Animate Parameter feature. This will enable the

Motion Controller.

You can graph the inverse by following these steps:

• From the Graph menu, choose New Function.

• From the Equation menu, choose x � f(y).

• Then, select parameter b and type ^y and click on OK.

• From the Graph menu, choose Plot Function.


Documents

Section 6.1 Page 310 to 322