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Inverse Functions
Definition of the Inverse Function
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Let f and g be two functions such that
f (g(x)) = x for every x in the domain of g
and g(f (x)) = x for every x in the domain of f.
The function g is the inverse of the function f, and denoted by f ­1 (read “f­inverse”). Thus, f ( f ­1
(x)) = x and f ­1( f (x)) = x. The domain of f is equal to the range of f ­1, and vice versa.
Text Example
Show that each function is the inverse of the other:
f (x) = 5x and g(x) = x/5.
Solution To show that f and g are inverses of each other, we must show that f (g(x)) = x and g( f (x)) = x. We begin with f (g(x)).
f (x) = 5x
f (g(x)) = 5g(x) = 5(x/5) = x.
Next, we find g(f (x)).
g(x) = 5/x
g(f (x)) = f (x)/5 = 5x/5 = x.
Notice how f ­1 undoes the change produced by f.
Finding the Inverse of a Function
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The equation for the inverse of a function f can be found as follows:
Replace f (x) by y in the equation for f (x).
Interchange x and y.
Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.
If f has an inverse function, replace y in step 3 with f ­1
(x). We can verify our result by showing that f ( f ­1(x)) = x and f ­1( f (x)) = x.
Text Example
Find the inverse of f (x) = 7x – 5.
Solution Step 1
Replace f (x) by y.
y = 7x – 5
Step 2 Interchange x and y.
Step 3 Solve for y.
Step 4 Replace y by f ­1(x).
x = 7y – 5
This is the inverse function.
x + 5 = 7y
Add 5 to both sides.
x + 5 = y
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Divide both sides by 7.
f ­1(x) =
x + 5
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Rename the function f ­1(x).
The Horizontal Line Test For Inverse Functions
• A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point.
Example
• Does f(x) = x2+3x­1 have an inverse function?
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­10 ­8 ­6 ­4 ­2
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8 10
Solution:
Example
This graph does not pass the horizontal line test, so f(x) = x2+3x­1 does not have an inverse function.
10
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­10 ­8 ­6 ­4 ­2
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­10
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8 10
Inverse Functions