Review of functions

A function is a particular relation. So, if f is a function from A into B then

1. f ⊏ A x B
   
   
2. for every a ∈ A, there is a unique b ∈ B such that (a,b) ∈ f.
   
   

In general, we denote f is a function from A to B as

    f: A →B

and we say that f maps A into B or f transforms A into B.

A function is also called a mapping or a correspondence.

If f: A →B is a mapping (function) and (a,b) ∈ f, then we write f(a)=b, f(a) is called the image of a.

A is called the domain of f  and  B is called the codomain of f.

The set f(A) of all images of A under the mapping f is called the

range of f.

Note that the range and the codomain are not always equal.

Consider the following examples.

1. Let A represent the set of articles in a supermarket and B the set of their prices. There is a correspondence between the articles and their prices.
   
   
2. Let X represent the set of schools in Texas and Y the set of their principals. There is a correspondence between each school and its principal.
   Let N={1,2,.........} be the set of natural numbers and S={1,8,27,....} be the set of the cubes of each natural number.
   
   
   

Example 1:

Suppose A = {1,2,3} B = {a,b,c,d}

A rule f is given by

    f(1) = b

    f(2) = c

    f(3) = a

    f(3) = d

Is f a function?

According to what we have learned about functions, for every
a ∈ A there exists a unique b ∈ B. But f(3) = a and
f(3) = d. Since 3 is assigned to two elements of B, f is not a function.

Example 2:

Let f: Z →Z where Z is the set of integers. We define f as

f(x) = 1 if x is even

f(x) = -1 if x is odd

f is a function.

The domain of f is Z

The codomain of f is Z

But the range of f = {-1, 1}

Example 3:

Which of the following are functions?

The relations f, g and q send every member of A into a unique member of B. So f, g & q are functions.

h is not a function since 10 corresponds to c as well as d and p is not a function because c is not assigned to any member of D.

Definition:

Let A and B be two non-empty sets. A function f from A to B, denoted by f: A→B, is a rule that assigns each member of A to a unique member of B.

A is called the domain of f

B is called the codomain of f and

f(A) is called the range of f

Or,

a relation is called a function if and only if no two different ordered pairs in the relation have the same first coordinate.

If f maps A into B, where A,B are two non-empty sets such that x ∈A is associated with y ∈B then y is called the f image of x or just the image of x, and is written as f(x)=y. Also, x is called the pre-image or inverse image of y.

Example 4:

Let A = {1,2,3} B = {4,5}

How many different functions can be had from A into B ?
List them all.

Solution:

There are eight possible functions from A into B.

Notice that n(A) = 3

                 n(B) = 2

number of functions = 23 = 8.
In general, if n(A) = m and n(B) = n then the number of possible mappings from A to B is nm.

Try these questions
1.	Show that f(x) + f(1/x) = 0.
	Solution:
2.	Let f: R →R defined by
	f(x) = x + 2 if x ≥3      = x - |x| if -3≤x <3      = 1 - x if x≤-3 Find f(2), f(5), f(-4), f(-1.6.....)
	Solution:
	Consider f (2) 2 < 3     ∴ f(x) = x - |x|     ∴ f(2) = 2 - |2|             = 2 - 2 = 0 Consider f (5)    5 > 3 use     f(x) = x+2           f(5) = 5+2 = 7 Consider f (-4)   -4 < -3 ∴ we use f(x) = 1-x           f(-4) = 1 - (-4)                 = 1 + 4                 = 5 Consider f(+1.6 . . . ) -3 < +1.6 . . < 3 We use f(x) = x - |x| f(+1.6 . . . ) = +1.6 . . . - |+1.6 . . . |                  = +1.6 . . . - [+(+1.6. . .)]                  = +1.6 . . . - 1.6 . . .                  = 0

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