# Inverse Functions

**I. Inverse Functions
**

If we form the composition of two functions , we should get

the identity function h (x) = x. So to speak, inverse

functions “undo” each other. Also note that if a function is

given as a set of ordered pairs , its inverse has all the x-coordinates

interchanged with their corresponding y-

coordinates .

**Ex:**f (x) = x + 2 and g (x) = x – 2 are inverses because

f (g (x)) = f (x - 2) = (x - 2) + 2 = x, and

g (f (x)) = g (x + 2) = (x + 2) – 2 = x.

The functions f and g are

**inverses**of each other iff:

1. f (g (x)) = x for every x in the domain of g, and

2. g (f (x)) = x for every x in the domain of f.

In this case we write g (x) as f

^{ -1}(x)

This is

**NOT**a negative exponent it is just the notation

**II. The Graph of an Inverse Function**

If we interchanged the x’ s and y’ s for every point on the

graph of a function , the graph would be reflected about

the line y = x.

**Ex:**f (x) = x – 1 and f

^{ -1}(x) = x + 1

**III. One-to-One Functions
**

Draw the graph of y = x

^{2}. Reflect it about the line y = x.

Is this inverse a function.

When is the inverse of a function a function?

This brings us to the

**Horizontal Line Test :**

• If every horizontal line meets the graph of a function

in at most one point, then the inverse of this function

will be a function.

• If no horizontal line intersects the graph of a function

in more than one point, no x- value is matched with

more than one y- value .

This brings us to the definition of a

**one-to-one function:**

• A function f is

**one-to-one**if each value of the

dependent variable corresponds to exactly one value

of the independent variable .

A function has an

**inverse iff**the function is

**one-to-one**.

**Ex:**Which of the following functions has an inverse

function?

(a) f (x) = |x|

(b) g (x) = x

^{3}

**IV. Finding Inverse Functions Algebraically
**

Steps for finding the inverse of a function:

If the function is one-to-one

1. Replace f (x) with y.

2. Interchange the roles of x and y.

3. Solve this new equation for y .

4. Replace y by f

^{-1}(x).

**Ex:**Find the inverse of each of the following functions.

(c) h (x) = x

^{2}for x ≥ 0

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