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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

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,


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) = 3x2 – 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.