3.4 Inverse Functions3.4 Inverse Functions

Goal: Find and use inverses of linear

and nonlinear functions.

Inverse FunctionsInverse FunctionsA function and its inverse can be

described as the "DO" and the "UNDO" functions. 

A function takes a starting value, performs some operation on this value, and creates an output answer. 

The inverse of this function takes the output answer, performs some operation on it, and arrives back at the original function's starting value.

Domain Range

This "DO" and "UNDO" process can This "DO" and "UNDO" process can be stated as a composition of be stated as a composition of functions. functions. If functions f and g are inverse

functions, f(g(x)) = g(f(x)) = x.

◦Example: If f(x) = x-1 and g(x) = x +1then f(g(x)) = x and g(f(x))

= x

Think of them as "undoing" one another and leaving you right where you started.

Definition of an InverseDefinition of an Inverse The inverse of a function is the set

of ordered pairs obtained by interchanging the first and second elements of each pair in the original function.        

Notation:  If  f  is a given function, then

f -1 denotes the inverse of  f.

Finding the Inverse of a Finding the Inverse of a FunctionFunctionBasically, the process of finding an

inverse is simply the swapping of the x and y coordinates. 

This newly formed inverse will be a relation, but may not necessarily be a function – perform horizontal line test of the given function for inverse existence!

One-to-one (1-1) functions are the functions that pass the horizontal line test. If a function fails the horizontal line test, restriction of its domain is required.

Three Methods to find the Three Methods to find the Inverse:Inverse:1. Swapping x and y-values

2. Solving Algebraically:a. Set the function = yb.  Swap the x and y variablesc.  Solve for y

3. Reflect graph over the line y = x.

Swapping x and y-valuesSwapping x and y-values

Given relation, find the inverse relation.

xx -2-2 -1-1 00 11 22

yy 44 22 00 -2-2 -4-4

Solving AlgebraicallySolving Algebraically

Find the equation of the inverse of the relation f(x) = 2x – 4.

1. Set the function = y y = 2x – 4

2. Swap the x and y variables x = 2y - 4

1. Solve for y y = (x + 4)/2

Reflect the graph over y = Reflect the graph over y = xx

Graph original function

f(x) = 2x + 3It is drawn in blueblue.

If reflected over the Identity line, y = x, the original

function becomes the redred

dotted graph. 

Practice:Practice: Verifying Inverse Verifying Inverse Functions!!!Functions!!!

1. Verify that f(x) = 2x – 4 and f-1(x) = ½x + 2 are inverses.

2. Verify that f and g are inverse 2. Verify that f and g are inverse functionsfunctions

f(x) = 3x – 1 and g(x) = ⅓x + ⅓