e - book for college algebra king fahd university of petroleum & minerals 2.3 e - book for...

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals

KFUPM - Prep Year Math Program (c) 20013 All Right Reserved

2.3


Functions

Definition of a Function Domain of a Function Range of a Function Function Notation Difference Quotient



Definition of a Function

In what follows A and B will denote two nonempty sets.

A function from set A to set B is a rule of correspondence that assigns (or maps) to each element x in set A exactly one element y in set B.

The set A is called the domain of the function and the subset of the set B consisting of all the outcomes of the function is called the range of the function.

The variables x of A and y of B are called the independent variable and dependent variable, respectively.



Example 1Find the domain and range for the function

The domain consists of all the numbers x which satisfy so . In terms of intervals, the domain can be written as . The range in this example is obvious from the definition of the function. On one hand, since the square root of a real number is greater than or equal to zero, an element y of the range must satisfy the inequality . In other words, in interval notations, the range of the function in this example is




The domain consists of all numbers of all numbers x except those numbers that makes equal zero. But if or or . Therefore, the domain in interval notation is

To determine the range in this case, we need to solve for x in order to determine the restriction that the function imposes on the numbers of the range.




To do this we perform the following steps

2 22

2

2

16 7 1 6 7 1 0

6 7

6 36 4 7 1

2

3 16

y y x x yx yx yx x

y y y yx

y

y y yx

y

ymust satisfy

From which we see that and Thus, the range is:

ymust satisf




The domain consists of all numbers x for which and . To find the domain, then we need to solve the quadratic inequality

which has solution Therefore, the domain is . Finding the range of this function requires the use of techniques of calculus, and therefore, is beyond the scope of this course.



Function Notations

A function that is represented by an expression in x defining y is usually denoted by the letter f (or g, h etc), while range element y corresponding to a domain element x is denoted by (or , etc) and is called the value of the function at (or the function value at x). For example for the function

we write

.



Example 4For compute and .

2

2

2

2

22

0 0 3 0 1

2 2 3 2 1 4 6 1

1 1 3 1 1

2

1

11

3 1

5 51 3 3 1

t t

t

f

f

f t

f t t

t t t

t

t



Example 5For compute and .

10 1

0 11 1

22 1 31

11 1

11 1 2

f

f

f tt

f tt t



Difference Quotients

The expressions or f a h f a

h

f x f a

x a

are called the difference quotients of the function f. These differences have important geometric meaning in relation to the “derivative” of the function f at the point a. In the following examples concentrate on the algebraic techniques used in calculating these quantities.



Example 6For compute for .

2 2

2

3 2 3 22

2 2

3 10

25

s

2

in2

5 2ce

x xf x f

x x

x x

xx x

xx

x



Example 7 Let and , which of the following correspondences represent a function from A to B ?

a) Does not represent function since the element g has no assignment.

b) Does not represent a function since there are two outputs for the input g.



Example 8 Let and , which of the following correspondences represent a function from A to B ?

Both (c) and (d) does represent functions from set A to B. Note that, in (d) both the inputs b and g are assigned to the same output u.

There is nothing in the definition of a function that prohibits two (or more) inputs to be assigned to the same output.

e - book for college algebra king fahd university of petroleum & minerals 2.3 e - book for...

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