Your Algebra Homework Can Now Be Easier Than Ever!

Inverse Functions


– Verify inverse functions
– Use the horizontal line test to determine if a
function is a one-to-one function.
– Find the inverse of a function.
– Given a graph, graph the inverse .
– Find the inverse of a function & graph both
functions simultaneously .

What is an inverse function?

• A function that “undoes” the original function.
• A function “wraps an x” and the inverse would
“unwrap the x” resulting in x when the 2
functions are composed on each other.

Given that f(x) = 7x − 2, use composition of
functions to show that f−1(x) = (x + 2)/7.

Do all functions have inverses?

• Yes, and no. Yes, they all will have inverses,
BUT we are only interested in the inverses if


• Recall, functions must pass the vertical line test
when graphed. If the inverse is to pass the
vertical line test , the original function must pass
the HORIZONTAL line test (be one-to-one)!

One-to-One Functions

A function f(x) is a one-to-one function if
x- values do not share the same y-values.

Remember that a function will have
different x -values.
A one-to-one function will have different
x- values and different y -values.

Why are one-to-one functions important?

One-to-One Functions
Inverse functions

Horizontal Line Test

• Use to determine whether a function is

• A function is one-to-one if and only if no
horizontal line intersects its graph more
than once

Horizontal-Line Test

Graph f(x) = −3x + 4.

Example: From
the graph at the
left, determine
whether the
function is one-to-
one and thus
has an inverse
that is a function.

From the
graph at the
left, determine
whether the
function is
and thus has
an inverse that
is a function.

How do you find an inverse?

• “Undo” the function.
• Replace the x with y and solve for y .

How to find the Inverse of a
One-to-One Function

1. Replace f(x) with y in the equation.
2. Interchange x and y in the equation.
3. Solve this equation for y .
4. Replace y with f-1(x).
Any restrictions on x or y should be considered and
included with the equation.

Remember: Domain and Range are interchanged
for inverses.


Determine whether the function f(x) = 3x − 2
is one-to-one, and if it is, find a formula for

How do their graphs compare ?

• The graph of a function and
its inverse always mirror
each other through the line

• Example:y = (1/3)x + 2 and
its inverse = 3(x-2)

• Every point on the graph
(x,y) exists on the inverse
as (y,x) (i.e. if (-6,0) is on
the graph, (0,-6) is on its

Graph of Inverse f-1 function

• The graph of f-1 is obtained by reflecting the
graph of f across the line y = x.

• To graph the inverse f-1 function:
Interchange the points on the graph of f to
obtain the points on the graph of f-1.


Graph f(x) = 3x − 2 and

using the same set of axes.
Then compare the two graphs .

Determine the domain and range of the function
and its inverse.


Properties of One -to-One Functions
and Inverses

• If a function is one-to-one, then its
inverse is a function.
• The domain of a one-to-one function f is
the range of the inverse f-1.
• The range of a one-to-one function f is
the domain of the inverse f-1.
• A function that is increasing over its
domain or is decreasing over its domain
is a one-to-one function.

Restricting a Domain

• When the inverse of a function is not a
function, the domain of the function can
be restricted to allow the inverse to be a
• In such cases, it is convenient to consider
“part” of the function by restricting the
domain of f(x). If the domain is restricted,
then its inverse is a function.

Restricting the Domain

Recall that if a function is not one-to-one,
then its inverse will not be a function.

If we restrict the domain values of f(x) to those greater
than or equal to zero , we see that f(x) is now one-to-one
and its inverse is now a function.


• For f(x) = x² - 1, x < 0:
a.) Find the equation for the inverse , f-1

b.) Find the domain and range for the function and its inverse.

Prev Next

Start solving your Algebra Problems in next 5 minutes!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:

OR is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for visitors -- if you order Algebra Helper by midnight of June 22nd you will pay only $39.99 instead of our regular price of $74.99 -- this is $35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from for a 1$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:

Algebra Helper can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)


Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:

OR is an authorized reseller
of goods provided by Sofmath
Check out our demo!
"It really helped me with my homework.  I was stuck on some problems and your software walked me step by step through the process..."
C. Sievert, KY
19179 Blanco #105-234
San Antonio, TX 78258
Phone: (512) 788-5675
Fax: (512) 519-1805

Home   : :   Features   : :   Demo   : :   FAQ   : :   Order

Copyright © 2004-2021, Algebra-Answer.Com.  All rights reserved.