E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
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2.3
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Functions
Definition of a Function Domain of a Function Range of a Function Function Notation Difference Quotient
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
Example 2Find the domain and range for the function
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
Example 2Find the domain and range for the function
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
Example 3Find the domain and range for the function
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
Example 5For compute and .
10 1
0 11 1
22 1 31
11 1
11 1 2
f
f
f tt
f tt t
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.