C. Finding Inverse Functions Using an Algebraic Method
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College Algebra Graphs & Models—
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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)
yϭ
xϩ1
2y
xϭ
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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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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Section 5.1 One-to-One and Inverse Functions
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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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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