PreCalculus Class Notes RP9 Inverse Functions
Inverse Operations
Add
Subtract
Multiply
Inverses?
Is a = b always equivalent to a 2 = b ?
Divide
Square Root
Square
See RP4 notes.
One-to-One Function
A function f is a one-to-one function if, for elements c and d in the domain of f, c ≠ d implies
f(c) ≠ f(d).
Horizontal Line Test
If every horizontal line intersects the graph of a function f at most once, then f is a one-to-one
function.
Are the following functions one-to-one?
y
y
y
4
4
4
3
3
3
3
2
2
2
2
1
1
1
x
−4
−3
−2
y
4
−1
1
2
3
1
x
4
−4
−3
−2
−1
1
2
3
x
4
−4
−3
−2
−1
1
2
3
4
x
−4
−3
−2
−1
1
−1
−1
−1
−1
−2
−2
−2
−2
−3
−3
−3
−3
−4
−4
−4
−4
2
3
4
Increasing, Decreasing and One-to-One Functions
If a continuous function f is increasing on its domain, then every horizontal line will intersect the
graph of f at most once. By the horizontal line test, f is a one-to-one function. Similarly, if a
continuous function g is only decreasing on its domain, then g is a one-to-one function.
y
y
y
4
4
4
3
3
3
2
2
2
1
1
1
x
−4
−3
−2
−1
1
2
3
4
x
−4
−3
−2
−1
1
2
3
4
x
−4
−3
−2
−1
1
−1
−1
−1
−2
−2
−2
−3
−3
−3
−4
−4
−4
2
3
4
Inverse Function
Let f be a one-to-one function. Then f –1 is the inverse function of f if f −1 ( f ( x ) ) = x for every x
in the domain of f and f ( f −1 ( x ) ) = x for every x in the domain of f –1.
Example
If f ( x ) = x3 then f −1 ( x ) = 3 x so f −1 ( f ( x ) ) = 3 x3 = x and f ( f −1 ( x ) ) =
Finding an Equation for f –1
To find a formula for f –1, perform the following steps.
STEP 1: Verify that f is a one-to-one function. If not, f -1 does not exist.
STEP 2: Switch x and y to obtain x = f -1(y).
STEP 3: Solve the equation for y, obtaining the equation y = f -1(x).
To verify f –1(x), show that f −1 ( f ( x ) ) = x and f ( f −1 ( x ) ) = x
Example
Let f be a one-to-one function given by f(x) = x2 – 3, x ≥ 0.
(a) Find a formula for f –1(x).
(b) Verify that your result from part (a) is correct.
Example
Find f −1 ( x ) for f ( x ) =
2x
x−5
( x)
3
3
=x.
Inverses Using Tables
In the table on the left, f computes the percentage of the U.S. population with 4 or more years of
college in year x. Write a numerical representation of f –1 in the table on the right.
x
f
1940
5
1970
11
2000
27
–2
2
x
f(x)
f –1
x
0
3
2
5
3
7
Given the table of values for f(x), evaluate the following:
f ( 2)
f −1 ( 2 )
f −1 ( 3)
Domains and Ranges of Inverse Functions
f ( x ) = x3 + 1
f ( x) = 3 x −1
f −1 ( 5 )
f −1 ( −2 )
f ( x ) = x 2 − 2, x ≥ 0
f ( x) = x + 2
y
y
y
4
4
4
3
3
3
2
2
2
1
1
1
x
−4
−3
−2
5
10
−1
1
2
3
4
x
−4
−3
−2
−1
1
2
3
4
x
−4
−3
−2
−1
1
−1
−1
−1
−2
−2
−2
−3
−3
−3
−4
−4
−4
2
Domain
Range
The domain of f equals the range of f –1.
The range of f equals the domain of f –1.
3
4
Graphs of Functions and Their Inverses
The graph of f –1 is a reflection of the graph of f across the line y = x.
For the graph of f(x) above, evaluate
f ( 2)
f −1 ( 2 )
f −1 ( 3)
f −1 ( −4 )
Example
Let f(x) = x3 + 2. Graph f .
Then sketch a graph of f –1.
Table
DrawInv Y1
Graph
f −1 ( 0 )