Your Algebra Homework Can Now Be Easier Than Ever!

Composite Functions;One-to-one Functions;Inverse Functions

Composite Functions; One-to- one Functions ;
Inverse Functions

L13 Composite Functions; One-to-one Functions;
Inverse Functions

A composite function (read as “f composed
with g”) is defined by

The domain of is the set of all real x in the
of g for which g(x) is in the domain of f .

Example: Show a diagram for the composite function

Similarly we define:

Example: Let and g(x) = x^2 − 2. Find:

(c) Find the composite functions and their domains



Example: Using the tables, find (1).
What is the value of ?

Example: Find functions f and g such that if

Example: An oil spill in the ocean as sumes a circular
shape with an expanding radius r given by
where t is the number of minutes after the measurements
are started and r is measured in meters.

(a) Find a formula that gives the area A of the circular
region as a function of time t.

(b) What is the area at the beginning? (t = )
(c) What is the area 3 minutes later? (t = )

Inverse Relations and Inverse Functions

Recall that a relation is a set of all ordered pairs (x, y),
where x is an element from the domain of the relation and
y is the corresponding element from the range.

Thus, the inverse relation we defined as the set of all
ordered pairs ( y, x).

Example: Find the inverse of the fol lowing relations .
Which of the relations are functions? De termine whether
the inverse relations are functions.


Note: Not for every function the inverse relation is a

The inverse of a function is a function itself if and only if
for each y in the range there is only one x in the domain.
In other words, no two ordered pairs have the same
second coordinates , that is, no horizontal line intersects
the graph at more than one point.

The functions for which the inverses are also functions
are called one-to-one.

Horizontal Line Test

If each horizontal line intersects the graph of a
function f in at most one point, then f is one-to-one.

Example: Use the Horizontal Line Test to determine
whether the function is one-to-one.

Note: A function which is increasing/decreasing on an
interval I is one-to-one on I.

Note: A quadratic function y = a(x − h)^2 + k (a ≠ 0)
is not one-to-one, but, when considered on the restricted
domain, for example, on interval [h,+∞), it is one-to-one.

Inverse Functions

Remember, that the inverse of a function f is also a
function if and only if f is one-to-one.

Let f be a one-to-one function. Then g is the
inverse function of f if
for all x in the domain of g;
for all x in the domain of f.

If g is the inverse function of f, then we write g as
f-1(x) and read: “f-inverse”.

Example: Determine whether the following functions are
inverses of each other:

Cancellation Rules for Inverses

The inverse functions undo each other with respect to
their compositions:

f-1( f (x)) = x for all x in the domain of f
f(f-1( y)) = y for all y in the domain of f-1

Equivalent Form of the Cancellation Rules:

f (x) = y


f-1(y) = x
(x in domain of f)   (y in the domain of f-1)

Note on the Domains and Ranges of the Inverses:

Domain of f-1 = Range of f
Range of f-1 = Domain of f

Graphing Inverses:

If the graph of f is the set of points (x, y), then the graph
of f-1 is the set of points ( y, x).
Since, points (x, y) and ( y, x) are symmetric with
respect to the line y = x


the graphs of f and f-1 are symmetric with respect to
the line y = x.

Example: Given the graph of
y = f (x). Draw the graph of
its inverse.

Finding the Inverse of a One-to-one Function f:

1. Write y = f (x).
2. Solve the equation for x: x = f-1( y)
3. Inter change x and y .
4. Give your answer in the form: y = f-1(x).
Note: Consider all restrictions on the variables .

Example: Find f-1(x) if it exists.

Finding the Inverse of a Domain-restricted Function:
Find the inverse of

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 April 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 © 2004-2018, Algebra-Answer.Com.  All rights reserved.