Inverse Functions
Definition of the Inverse Function
•
•
•
•
•
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 “f-inverse”). 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.
1
Finding the Inverse of a Function
•
1.
2.
3.
4.
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.
Step 2
Interchange x and y.
Step 3
Solve for y.
y = 7x – 5
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
7
Divide both sides by 7.
f -1(x) =
x+5
7
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.
2
Example
• Does f(x) = x2+3x-1 have an inverse
function?
10
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
Example
Solution:
This graph does not pass the horizontal line
test, so f(x) = x2+3x-1 does not have an
inverse function.
10
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
Inverse Functions
3