Functions Worksheet With Answer Key Page 32
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46100
Chapter Two FUNCTIONS
Finding a Formula for the Inverse Function
In the next example, we find the formula for an inverse function.
Example 6
The cricket function, which gives temperature, T , in terms of chirp rate, R, is
1
T = f (R) =
· R + 40.
4
1
Find a formula for the inverse function, R = f
(T ).
Solution
The inverse function gives the chirp rate in terms of the temperature, so we solve the following
equation for R:
1
T =
· R + 40,
4
giving
1
T
40 =
· R
4
R = 4(T
40).
1
Thus, R = f
(T ) = 4(T
40).
Domain and Range of an Inverse Function
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The input values of the inverse function f
are the output values of the function f . Thus, the
1
1
domain of f
is the range of f . For the cricket function, T = f (R) =
R + 40, if a realistic
4
1
domain is 0 ≤ R ≤ 160, then the range of f is 40 ≤ T ≤ 80. The domain of f
is then
40 ≤ T ≤ 80 and the range is 0 ≤ R ≤ 160. See Figure 2.32.
T
◦
(
F)
1
f
80
✻
f
Range of
1
=
f
Domain of
f
❄
40
✛
✲
R
(chirps/min)
160
f
Domain of
1
=
f
Range of
:
Figure 2.32
Domain and range of an inverse function
Relation Between Composition and Inverses: A Function and Its Inverse Undo Each Other
Composing a function with its inverse gives a striking result, as we see in the next example.
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1
Example 7
Calculate the composite functions f
(f (R)) and f (f
(T )) for the cricket example. Interpret the
results.
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1
Solution
Since f (R) =
R + 40. and f
(T ) = 4(T
40), we have
4
1
1
1
1
f
(f (R)) = f
· R + 40 = 4
· R + 40
40 = R.
4
4
1
1
f (f
(T )) = f (4(T
40)) =
(4(T
40)) + 40 = T.
4
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