# Inverse Functions

**4.1 Inverse Functions
I. Basic Concepts**

A. Relation: a correspondence between two sets - a set of inputs called the domain & a set

of outputs called the range.

B. Function: a special type of relation where every input has one and only one output.

A function will pass a

**Vertical Line Test.**

C. One-to-one Function: a special type of function where every input has one & only one

output and every output has one & only one input. 1-1 functions pass a Vertical Line Test

and a

**Horizontal Line Test.**

D. The Inverse of a Function

Informal Definition: The inverse of a function undoes what the original function did.

Formal Definition: Functions f and g are inverse functions if and only if

(f o g)( x) = f (g (x)) = x and (g o f) (x) = g (f (x)) = x.

Caution: f inverse does not equal 1 over f

II. Finding the Formula for the Inverse of a Function

II. Finding the Formula for the Inverse of a Function

Before you do anything else, determine if the given function is 1-1 by making an accurate “quick

sketch” and then doing a VLT and a HLT. If the function is not 1-1, it does not have an inverse.

If it is 1-1, find its inverse.

A. Finding an Inverse Algebraically

1. Replace f (x) with y.

2. Interchange x and y.

3. Solve for y.

4. Replace y with f

^{-1}(x).

**Example 1**

a. Sketch the graph of and use the graph to determine whether the function is 1-to-1.

Since the graph of f (x) passes both a vertical line test

and a horizontal line test, it is a 1-to-1 function and

it does have an inverse.

b. If the function is 1-to-1, algebraically find a formula for the inverse.

B. Finding an Inverse Informally

To undo what the original function did, the inverse function must do the
opposite

operations in the opposite order.

**Example 2
**

Find the inverse of g (x) = 7x + 2 by thinking about the operations of the function and then undoing them.

Check your work algebraically.

Since the graph of g (x) is an oblique line, it passes both a vertical and a horizontal line test. Therefore g

does have an inverse.

Function g first multiplies x by 7 and second adds 2. To undo this, the inverse function must

first subtract 2 and second divide by 7. Therefore,

Check:

**III. Characteristics of Inverse Functions**

A. Only one-to-one functions have inverses.

B. The domain of f is the range of f ^{-1} and the range of f is the domain of f ^{-1}.

C. If you make a t-chart for f and then use those y values as the x values to
make a t-chart

for f ^{-1}, the two t-charts will be mirror images of one another, i.e., their x
values and y

values will be reversed.

D. f and f ^{-1} are symmetric around the y = x line.

E. If f and g are inverse functions, (f o g) (x) = f (g (x)) = x and (g o f) (x) =
g (f( x)) = x.

**Example 3**

Algebraically find the inverse of the given one-to-one function, f (x) = 3x^{2} – 5,
x ≥ 0.

Give the domain and range of f and f ^{-1}. Then graph f and f ^{-1} on the same set
of axes.

Note: Without the restriction on the domain, f would not be a 1-to-1 function.

Note: Because of the restriction on the domain of f, x ≥ 0, f ^{-1} has a
restriction

on its range, y ≥ 0, that is how we know the square root is positive and

not negative.

Domain of f = Range of f ^{-1} = [0, ∞ ) and Range of f =
Domain of f^{ -1} = [–5, ∞ )

**IV. Verifying Algebraically that Two Functions are
Inverses**

If (f o f ^{-1} )(x) = f (f^{ -1}(x)) = x and (f ^{-1} o f )(x)= f^{ -1}(f(x)) = x, then f and
f^{ -1} are inverse functions

**Example 4
**

For the function f, use composition of functions to show that f

^{-1}is as given.

Since f (f^{ -1}(x)) = x and f^{ -1}(f(x)) = x, f and f^{ -1} are
inverse functions.