Name: ___________________

Date:___________________

kwizNET Subscribers, please login to turn off the Ads!
Email us to get an instant 20% discount on highly effective K-12 Math & English kwizNET Programs!

High School Mathematics - 2
4.7 Inverse of Functions

Consider the function of of the set of children to the set of their mothers. It is obviously a many-one type function as several children may have the same mother. Its inverse however will not exist. In this case we will have a relation in which several ordered pairs of the form (mother, child) may have the same first component.

Consider the functions f1, f2, f3, f4, f5 exhibited by the following diagrams:

  1. f1(one-one into type), f1-1 doesn't exist.
  2. f2(one-one onto type), f2-1 doesn't exist.
  3. f3(many-one into type), f3-1 doesn't exist.
  4. f4(many-one onto type), f4-1 doesn't exist.
  5. f5(constant type), f5-1 doesn't exist.
The above illustrations lead to a conclusion that
A function f: A -> B will have its inverse g: B -> A if and only if
  1. f is a one-one function.
  2. f is an onto function i.e f is bijective.
Definition: If f is a function f: A - >B, then there will exist a function g: B ->A if f is a one-one onto function, and such that the range of f is the domain of g and domain of f is the range of g; then g is called the inverse of f and is denoted by f-1. Also f is called the inverse of g and is denoted as g-1.
Symbolically, a function f: x - >y , then its inverse is represented as f-1: y - > x, or if y = f(x), then its inverse is represented as f-1(y) = x.

Example: If f: R -> R be define by f(x) = x3 + 7, find a formula that defines f-1.
Solution: Method to find the inverse of a bijective function

  1. Step 1: Put f(x) = y, where y B and x A.
  2. Step 2: Solve f(x) = y to obtainx in terms of y.
  3. Step 3: In the relation obtained in step 2, replace x by f-1 to obtain the inverse of f.
  4. Step 4: Let f(x) = x3 + 7 = y, then x3 = y - 7
  5. Step 5: x = (y-7)1/3 = >f-1(y) = (y - 7)1/3
Answer: Hence f-1 : R -> R : f-1(x) = (x-7)1/3for all x R.


Directions: Solve the following problems. Also write at least 5 examples of your own.
Q 1: If f : R+ -> R such that f(x) = log8x,find f-1(x).
8y
log8y
8x

Q 2: f: R -> R be defined by f(x) = 10x - 7. if g = f-1 then g(x) is
1/(10x - 7)
(x + 7)/10
1/(10x + 7)

Q 3: Find the range of the function h(x) = x2 + 1
Range doesn't exist
Range = Domain
{(y: y is real,y>=1}

Q 4: If f: R=>R such that f(x) = e3x+2,find f-1e2
0
5
1

Q 5: Let f : A -> B, find f-1(B).
Inverse doesn't exist
B
A

Q 6: If f: R ->R is such that f(x) = log3x, f-1 is equal to
log x3
31/x
3x

Q 7: If f: R ->R be defined by f(x) = 3x- 4, then f-1(x) is
(x+4)/3
3x + 4
x/3 - 4

Q 8: If f:R -> R, write the inverse of f(x) = 3x + y.
(x-4)/3
(y-4)/3
-(3x+y)

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


Subscription to kwizNET Learning System costs less than $1 per month & offers the following benefits:

  • Unrestricted access to grade appropriate lessons, quizzes, & printable worksheets
  • Instant scoring of online quizzes
  • Progress tracking and award certificates to keep your student motivated
  • Unlimited practice with auto-generated 'WIZ MATH' quizzes
  • Child-friendly website with no advertisements


© 2003-2007 kwizNET Learning System LLC. All rights reserved. This material may not be reproduced, displayed, modified or distributed without the express prior written permission of the copyright holder. For permission, contact info@kwizNET.com
For unlimited printable worksheets & more, go to http://www.kwizNET.com.