# Inverse Functions

## One-to- One Functions and Their Inverses

Let f be a function with domain A. f is said to be one-to-one if no two
elements in A have the same image.

Example 1: De termine if the fol lowing function is one-to-one.

•A one-to-one function has an inverse function.
•The inverse function reverses whatever the first function did.

Example: The formula

is used to convert from x degrees Celsius to y degrees Fahrenheit. The formula

is used to convert from x degrees Fahrenheit to y degrees Celsius

•The inverse of a function f is denoted by f-1 , read “f-inverse”.

Example 2: As sume that the domain of f is all real numbers and that f is one -to-one. If

f(7) = 9 and f(9) = -12, then what is

Example 3:
If f and g are inverse functions, f (-2) = 3 and f (3) = -2 . Find g(-2) .

Domain and Range:

The domain of f is the range of f-1 and the range of f is the domain of f-1 .

These two statements mean exactly the same thing:

1. f is one-to-one (1-1)
2. f has an inverse function

Property of Inverse Functions

Let f and g be two functions such that ( f o g)(x) = x for every x in the domain of
g and (g o f )(x) = x for every x in the domain of f then f and g are inverses of
each other.

Example 4:
Show that the following functions are inverses of each other

How to find the inverse of a function: (if it exists!)

1. Replace “ f (x) ” by “y”.
2. Ex change x and y .
3. Solve for y .
4. Replace “y” by “ f-1(x) ”.
5. Verify!

Example 5:
Find the inverse function of f (x) = 2x - 7 .

Example 6:
Assume f (x) is a one-to-one function. Find the inverse function f-1(x) given that

Example 7:
Find the inverse function f-1(x) given that

Example 8:
Assume g(x) is a one-to-one function. Find the inverse function g-1(x) given that

Example 9: Assume g(x) is a one-to-one function. Find the inverse function g-1(x) given that

 Prev Next

Start solving your Algebra Problems in next 5 minutes!

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of July 21st 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 tutor.com 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...)

ORDER NOW!