Name: ___________________Date:___________________

 Email us to get an instant 20% discount on highly effective K-12 Math & English kwizNET Programs!

### High School Mathematics - 24.8 Types of Functions - (One To One)

 One-One Function A function f: A->B is called a one-one function if distinct elements of A have distinct images in B, i.e if a1, a2 €A and a1 ≠a2=>f(a1) ≠ f(a2) Equivalently, we say f:A -> B is one-one if and only if for all a1, a2 € A, f(a1) = f(a2) => a1 = a2. Note 1: A one-one function is also called an injective function or injection. Note 2: If A and B are finite sets and f: A -> B is injective. Then n(A)<= n(B). Note 3: If n(A) = P and n(B) = q, then the number of possible mappings from A to B is qp. Illustrations: If A = {4, 5, 6} and B = {a, b, c, d} and if A -> B such that f = {(4,a), (5,b), (6,c)}, then f is one-one. The mapping f: R->R such that f(x) = x2 is not a one-one function since f(-2) = 4 and f(2) = 4, that is two distinct elements -2 and 2 have the same image 4. Example: Find if the following functions are one-one or not. f: R->R, defined by f(x) = x3, x € R. Example:Find whether the following functions are one-one or not. f: R->R , defined by f(x) = x3, x € R f: Z->Z, defined by f(x) = x2 + 5 for all x € Z. Method to check the injectivity of a function Step 1: Take two arbitrary elements x and y in the domain of f. Step 2: Put f(x) = x Step 3: Solve f(x) = f(y). If it yields x = y only then f: A->B is a one-one function or injection. Remark: Let f: A ->B and let x,y € A. Then x = y=>f(x) = f(y) is always true from the definition, but f(x) = f(y)=>x = y is true only when f is one-one. Solution: 1. Let x, y be two arbitrary elements of domain f, (x,y € R) such that f(x) = f(y). Then f(x) = f(y) =>x3 = y3 =>x = y 2. Let x, y be two arbitrary elements of Z such that f(x) = f(y). Then f(x) = f(y) =>x2 +5 = y5 + 5 =>x2 = y2 =>x = + or - y. Since f(x) = f(y) does ot yield a unique answer and x = y but gives x = + or - y, so f is not a one-one function. Suppose we have f(2) and f(-2), for either cases we get 9, thus two distinct elements 2 and -2 have the same image. Hence f is one-one function. Example: A one to one function is a function in which every element in the range of the function corresponds with one and only one element in the domain. Example of a one-to-one function: { (0,1) , (5,2), (6,4) } Domain: 0, 5, 6 Range: 1,2, 4 Each element in the domain (0, 5, and 6) correspond with a unique element in the range. Therefore this function is a one-to-one function. Directions: Solve the following problems. Also write at least 5 examples of your own.
 Q 1: Determine if the given function is one-one. To each person on earth assign the number which corresponds to his age.YesNo Q 2: To each country in the world, assign the latitude and longitude of its capital. Determine if it is one-one or not.NoYes Q 3: To each book, written by only one author, assign the author. Determine if it is one-one or not.Noyes Q 4: To each country in the world which has a prime minister assign its prime minister.NoYes Q 5: {(x,y): y greater than or equal to x-3}, x € z is a function or not, YesNo Q 6: {(x, y) : y = 5x - 6 }.Is it a function?NoYes Q 7: {(x, y) : y = 5x - 6 }. If it is a function, what function is it?OntoOne-oneNot a function Q 8: {(x, y) : y = 4 for all values of x}, is it a function?yesno 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

© 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.