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C. Finding Inverse Functions Using an Algebraic Method

C. Finding Inverse Functions Using an Algebraic Method

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College Algebra Graphs & Models—



5–5



Section 5.1 One-to-One and Inverse Functions



WORTHY OF NOTE



Finding an Inverse Function



If a function is not one-to-one,

no inverse function exists since

interchanging the x- and

y-coordinates will result in a

nonfunction. For instance, interchanging the coordinates of (Ϫ2, 4)

and (2, 4) from y ϭ x2 results in

(4, Ϫ2) and (4, 2), and we have one xvalue being mapped to two y-values,

in violation of the function definition.



EXAMPLE 3



Solution



483











1. Use y instead of f (x).

2. Interchange x and y.



3. Solve the new equation for y.

4. The result gives the inverse function:

substitute f Ϫ1 1x2 for y.



In this process, it might seem like we’re using the same y to represent two different

functions. To see why there is actually no contradiction, see Exercise 97.



Finding Inverse Functions Algebraically

State the domain and range of the function given, then use the algebraic method to

find the inverse function, and state its domain and range.

2x

3

a. f 1x2 ϭ 1 x ϩ 5

b. g1x2 ϭ

xϩ1

3

a.

f 1x2 ϭ 1

x ϩ 5, x ʦ ‫ޒ‬, y ʦ ‫ޒ‬

3

yϭ 1

xϩ5

use y instead of f (x)

3

interchange x and y

x ϭ 1y ϩ 5

cube both sides

x3 ϭ y ϩ 5

solve for y

x3 Ϫ 5 ϭ y

x3 Ϫ 5 ϭ f Ϫ1 1x2

the result is f Ϫ1 1x2

f Ϫ1 1x2 ϭ x3 Ϫ 5, x ʦ ‫ޒ‬, y ʦ ‫ޒ‬

2x

b.

, x Ϫ1, y 2

g1x2 ϭ

xϩ1

2x

use y instead of f (x)



xϩ1

2y



interchange x and y

yϩ1

xy ϩ x ϭ 2y

multiply by y ϩ 1 and distribute

gather terms with y

x ϭ 2y Ϫ xy

factor

x ϭ y12 Ϫ x2

x

ϭy

solve for y

2Ϫx

x

gϪ1 1x2 ϭ

, x 2, y Ϫ1

2Ϫx

Now try Exercises 39 through 46







In cases where a given function is not one-to-one, we can sometimes restrict the

domain to create a function that is, and then determine an inverse. The restriction

we use is arbitrary, and only requires that the result produce all possible range

values. Most often, we simply choose a limited domain that seems convenient or

reasonable.

EXAMPLE 4







Restricting the Domain to Create a One-to-One Function



Given f 1x2 ϭ 1x Ϫ 42 2, restrict the domain to create a one-to-one function, then

find f Ϫ1 1x2 . State the domain and range of both resulting functions.



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CHAPTER 5 Exponential and Logarithmic Functions



Solution







The graph of f is a parabola, opening upward with the

vertex at (4, 0). Restricting the domain to x Ն 4

(see figure) leaves only the “right branch” of the

parabola, creating a one-to-one function without

affecting the range, y ʦ 30, q 2 . For f 1x2 ϭ 1x Ϫ 42 2

with restricted domain x Ն 4, we have

f 1x2 ϭ 1x Ϫ 42 2

y ϭ 1x Ϫ 42 2

x ϭ 1y Ϫ 42 2

Ϯ 1x ϭ y Ϫ 4

1x ϩ 4 ϭ y



y



5



given function

use y instead of f (x)



x



5



interchange x and y



Ϫ2



take square roots

solve for y, use 1x since x Ն 4



The result shows f 1x2 ϭ 1x ϩ 4, with domain x ʦ 3 0, q 2 and range

y ʦ 34, q 2 (the domain of f becomes the range of f Ϫ1, and the range of f becomes

the domain of f Ϫ1).

Ϫ1



Now try Exercises 47 through 52







We can further illustrate the ideas from Example 4 using a calculator’s ability to draw

inverses. On the Y= screen, set Y1 ϭ X2 and GRAPH this function on a

size of 3Ϫ7.5, 7.54 for x and 3Ϫ3, 7 4 for y. Then go to the home screen and access

the DrawInv feature using 2nd PRGM (DRAW) 8:DrawInv. This will place the

DrawInv feature on the home screen, where we specify that we want the inverse of Y1

(Figure 5.7). Pressing

returns us to the graph, where we discover that since

y ϭ x2 is not one-to-one, the calculator has graphed the inverse relation (Figure 5.8).

WINDOW



ENTER



Figure 5.8



Figure 5.7



7



Ϫ7.5



7.5



Ϫ3



Returning to the Y= screen and restricting the domain of Y1 to x Ն 0 (Figure 5.9), then

repeating the sequence above, produces the graphs shown in Figure 5.10. Note the given

function is now one-to-one, and its inverse is also now a function (and one-to one).

Figure 5.10



Figure 5.9



7



Ϫ7.5



7.5



Ϫ3



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College Algebra Graphs & Models—



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Section 5.1 One-to-One and Inverse Functions



485



While we now have the ability to find the inverse of a function, we still lack a definitive method of verifying the inverse is correct. Actually, the diagrams in Figures 5.5

and 5.6 suggest just such a method. If we use the function f itself as an input for f Ϫ1,

or the function f Ϫ1 as an input for f, the end result should simply be x, as each function

“undoes” the operations of the other. From Section 3.6 this is called a composition of

functions and using the notation for composition we have,

Verifying Inverse Functions

If f is a one-to-one function, then the function f Ϫ1 exists, where

1 f ‫ ؠ‬f Ϫ1 2 1x2 ϭ x



1 f Ϫ1 ‫ ؠ‬f 2 1x2 ϭ x



and



EXAMPLE 5







Finding and Verifying an Inverse Function



Solution







Since the graph of f is the graph of y ϭ 1x shifted 2 units left, we know f is oneto-one with domain x ʦ 3 Ϫ2, q 2 and range y ʦ 30, q 2. This is important since the

domain and range values will be interchanged for the inverse function. The domain

of f Ϫ1 will be x ʦ 30, q 2 and its range y ʦ 3Ϫ2, q 2.



Use the algebraic method to find the inverse function for f 1x2 ϭ 1x ϩ 2. Then

verify the inverse you found is correct.



f 1x2 ϭ 1x ϩ 2

y ϭ 1x ϩ 2

x ϭ 1y ϩ 2

x2 ϭ y ϩ 2

x2 Ϫ 2 ϭ y

f Ϫ1 1x2 ϭ x2 Ϫ 2



Verify







C. You’ve just seen how we

can find inverse functions using

an algebraic method



1 f ‫ ؠ‬f Ϫ1 2 1x2 ϭ

ϭ

ϭ

ϭ

ϭ



f 3 f Ϫ1 1x2 4

2f Ϫ1 1x2 ϩ 2

21x2 Ϫ 22 ϩ 2

2x2

x✓



1 f Ϫ1 ‫ ؠ‬f 2 1x2 ϭ f Ϫ1 3 f 1x2 4

ϭ 3 f 1x2 4 2 Ϫ 2

ϭ 3 2x ϩ 24 2 Ϫ 2

ϭxϩ2Ϫ2

ϭx✓



given function; x Ն Ϫ2

use y instead of f(x)

interchange x and y

solve for y (square both sides)

subtract 2

the result is f Ϫ1 1x2 ; D: x ʦ 30, q 2, R: y ʦ 3Ϫ2, q2

f Ϫ1 1x2 is an input for f



f adds 2 to inputs, then takes the square root

substitute x 2 Ϫ 2 for f Ϫ1 1x2

simplify

since the domain of f Ϫ1 1x2 is x ʦ 3 0, q )



f (x) is an input for f Ϫ1

f Ϫ1 squares inputs, then subtracts 2

substitute 1x ϩ 2 for f (x)

simplify

result



Now try Exercises 53 through 78







D. The Graph of a Function and Its Inverse

Graphing a function and its inverse on the same axes reveals an interesting and useful

relationship—the graphs are reflections across the line y ϭ x (the identity function).

xϪ3

1

3

ϭ x Ϫ . In

Consider the function f 1x2 ϭ 2x ϩ 3, and its inverse f Ϫ1 1x2 ϭ

2

2

2

Figure 5.11, the points (1, 5), (0, 3), (Ϫ32, 0), and (Ϫ4, Ϫ5) from f (see Table 5.3) are

graphed in blue, with the points (5, 1), (3, 0), (0, Ϫ32), and (Ϫ5, Ϫ4) (see Table 5.4) from

f Ϫ1 graphed in red (note the x- and y-values are reversed). Graphing both lines illustrates this symmetry (Figure 5.12).



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College Algebra Graphs & Models—



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CHAPTER 5 Exponential and Logarithmic Functions



Table 5.3



Figure 5.11



Figure 5.12



y



x



f(x)



1



5



0



3



3

Ϫ

2



0



Ϫ4



Ϫ5



EXAMPLE 6



5



yϭx



3



3



Ϫ2







f(x) ϭ 2x ϩ 3



1



Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

Ϫ1



yϭx



4



2



1



2



3



4



5



3

fϪ1(x) ϭ x Ϫ

2



x



f Ϫ1(x)



x



5



4



f(x) ϭ 2x ϩ 3



Table 5.4



y



2



5



1



3



0



1



Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

Ϫ1

Ϫ2



Ϫ3



Ϫ3



Ϫ4



Ϫ4



Ϫ5



Ϫ5



1



2



3



fϪ1(x) ϭ



4



5



Ϫ



0



x



xϪ3

2



Ϫ5



3

2



Ϫ4



Graphing a Function and Its Inverse

Given the graph shown in Figure 5.13, draw a graph of the inverse function.

Figure 5.13

y

5



Figure 5.14

y



f(x)



5



f(x)



(2, 4)

f Ϫ1(x)

(1, 2)

(0, 1)

Ϫ5



5



x



Ϫ5



Ϫ5



Ϫ5



Solution







(4, 2)

(2, 1)

5 x

(1, 0)



From the graph, the domain of f appears to be x ʦ ‫ ޒ‬and the range is y ʦ 10, q 2.

This means the domain of f Ϫ1 will be x ʦ 10, q 2 and the range will be y ʦ ‫ޒ‬. To

sketch f Ϫ1, draw the line y ϭ x, interchange the x- and y-coordinates of the

selected points, then plot these points and draw a smooth curve using the domain

and range boundaries as a guide. The result is shown in Figure 5.14.

Now try Exercises 79 through 84







A summary of important concepts is provided here.

Functions and Inverse Functions

1. If the graph of a function passes the horizontal line test, the function is

one-to-one.

2. If a function f is one-to-one, the function f Ϫ1 exists.

3. The domain of f is the range of f Ϫ1, and the range of f is the domain of f Ϫ1.

4. For a function f and its inverse f Ϫ1, 1 f ‫ ؠ‬f Ϫ1 2 1x2 ϭ x and 1 f Ϫ1 ‫ ؠ‬f 21x2 ϭ x.

5. The graphs of f and f Ϫ1 are symmetric to the line y ϭ x.

These ideas can be illustrated and reaffirmed using a graphing calculator. To begin,

X3

3

enter the functions Y1 ϭ 2 1

X Ϫ 2 and Y2 ϭ

ϩ 2 (which appear to be inverse

8

functions) on the Y= screen, along with Y3 ϭ X. Press ZOOM 6:ZStandard, then

ZOOM

5:ZSquare to obtain a screen that is in perspective. The graphs seem to be

reflections across the line y ϭ x (Figure 5.15). To verify, we can use the TABLE



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