Section 1.5
Inverse Functions
A function is one-to-one (1-1) if no two elements in the domain have the same
image. If a function is 1-1 then it has an inverse.
The inverse of a function f is denoted by f −1 . Note that f −1 ( x) ≠
1
.
f ( x)
Given the graph of a function, we can determine if that function has an
inverse function by applying the Horizontal Line Test.
The Horizontal Line Test
−1
A function f has an inverse function, f , if there is no horizontal line that
intersects the graph in more than one point. Means that each x values have an
unique y- value.
Example 1:
a. Is this function one-to-one?
,−
,
, − ,
,
,
, − ,
, − ,
,
Example 2: Which of the following are one-to-one functions?
y
1
x
−2
−1
1
2
−1
a.
Section 1.5 – Inverse Functions
1
b. f ( x ) =
1
x
c. g( x ) = x − 1 + 2
Domain and Range
The inverse function reverses whatever the first function did; therefore, the
domain of f is the range of f −1 and the range of f is the domain of f −1 .
Example 3: Suppose that f and g are inverses. If
− =
and
= . Find
.
=− ,
= ,
Example 4: Suppose that f (3) = 7 , f (7) = 2 , g (7) = 4 , and g (3) = 0 . Find
f
−1
( g −1 (4)).
Property of Inverse Functions
Let f and g be two functions such that (f o g )( x ) = x for every x in the domain
of g and ( g o f )( x ) = x for every x in the domain of f then f and g are inverses
of each other.
Section 1.5 – Inverse Functions
2
Example 3: Determine whether the following functions are inverses of each
other.
x+2
f ( x ) = −3x − 2 and g( x ) =
3
How to find the inverse of a one-to-one function:
1.
2.
3.
4.
5.
Replace f(x) by y.
Exchange x and y.
Solve for y.
Replace y by f −1 ( x ).
Verify.
Example 4: Find an equation for, f −1 ( x) , the inverse function of the following
one-to-one function.
a. Suppose f is defined for x > 3, by f (x) = ( x − 3) 2 .
Section 1.5 – Inverse Functions
3
=−
b.
+ , Find f −1 ( x) .
− 3x + 1
Then find the range of f −1 ( x) .
2x − 5
c. f ( x ) = −
Example 5: Find the linear function f if
Section 1.5 – Inverse Functions
=− −
=
4
Example 6: Here is the graph of a function and the inverse of that function. If
a point (a, b) is on the graph of a function then the point (b, a) is on the graph
of the inverse of that function.
Section 1.5 – Inverse Functions
5