Your Algebra Homework Can Now Be Easier Than Ever!

One-to-one, Onto, and Inverse Functions

• In this section, we will look at three special
classes of functions and see how their properties
lead us to the theory of counting.

• So far, we have the general notion of a function
f: X → Y, but in terms of the comparative sizes
of the three sets involved (X, Y and f ), all we
can say is that |f | = |X|.

• In this section, we compare |X| with |Y|.

One-to- one Functions

• Definition: A one-to-one (injective) function f
from set X to set Y is a function such that each
x in X is related to a different y in Y .

• More formally, we can restate this definition as

 f :X → Y is 1-1 provided
f(x1) = f(x1) implies x1 = x2,

or f :
X → Y is 1-1 provided
x1 ≠ x2 implies f(x1) ≠ f(x2).

Illustrative Examples

• The function be low is 1-1:

This function is not:

Proving Functions Are 1-1

• If f: RR is given by f (x) = 3x + 7, prove it is

• Proof: As sume f (a) = f (b). Show a = b.

Now f (a) = f (b) means 3a + 7 = 3b + 7, so
3a = 3b, therefore a = b.

• Why is f: RR given by f (x) = x2 not 1-1?

• Since 9 = f(3) = f(-3), but 3 ≠ -3, the definition is

Onto Functions

• Definition: A function f: X → Y is said to be onto
(surjective) if for every y in Y, there is an x in X
such that f (x) = y.

• This can be restated as: A function is onto when
its image equals its range, i.e. f (X) = Y.

• Examples:



Testing Onto For Infinite Functions

• Show that f: RR given by f (x) = 5x - 7 is onto.

• Let y be in R. Then (y + 7) and (y + 7)/5 are also
real numbers .

Now f( (y + 7)/5 ) = 5[(y + 7)/5] - 7 = y, hence
if y is in R, there exists an x in R such that
f (x) = y.

• Is f: RR given by f (x) = 1/x onto?

• No! There is no x in R that has output = 0.

One-to-one Correspondences

• Definition: A function is called a one-to-one
correspondence (bijection) if it is one-to-one and

• One-to-one correspondences define the theory of
counting. Why?

• If f: X → Y is one-to-one, then |X| ≤ |Y|, and if f
is onto, then |X|≥ |Y|, so if f is both, |X| = |Y|.

• Hence, to count the elements of an unknown set ,
we create a 1-1 correspondence between the set
and a set of known size. Simple !

Inverse Functions

• Recall that the inverse relation is created by
inverting all the ordered pairs that comprise the
original relation.

• When is the inverse of a function itself a

not onto
(f -1 not def.)

not 1-1
(f -1 not well-def.)

(f -1 is a function)

Finding Inverse Functions

• Theorem: If f: X → Y is a one-to-one and onto,
then f -1 is a one-to-one and onto function.

• Given f , how do we find f -1?

• Let f: R → R be given by f (x) = 4x - 1 = y. Now,
swap x and y and solve for y :

• Thus f -1 (x) = (x + 1)/4.

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 December 11th 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 © 2004-2017, Algebra-Answer.Com.  All rights reserved.