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High School Mathematics - 2
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.

  1. f: R->R, defined by f(x) = x3, x R. Example:Find whether the following functions are one-one or not.
    1. f: R->R , defined by f(x) = x3, x R
    2. f: Z->Z, defined by f(x) = x2 + 5 for all x Z.
Method to check the injectivity of a function
  1. Step 1: Take two arbitrary elements x and y in the domain of f.
  2. Step 2: Put f(x) = x
  3. 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.

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.

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