College AlgebraFunctions and Graphs

(Chapter1)L:8

1

Instructor: Eng. Ahmed abo absa

University of PalestineIT-College

2

Objectives

Chapter1

Cover the topics in Section ( 1-3):Functions

After completing this section, you should be able to:

1. Know what a relation, function, domain and range are. 2. Find the domain and range of a relation. 3. Identify if a relation is a function or not. 4. Evaluate functional values.

3

Functions

Chapter1

Relation

A relation is a set of ordered pairs where the first components of the ordered pairs are the input values and the second components are the output

values.A relation is a rule of correspondence that relates two sets.

For instance, the formula I = 500r describes a relation between the amount of interest I earned in one year and the interest rate r.

In mathematics, relations are represented by sets of ordered pairs (x, y) .Function

A function is a relation that assigns to each input number EXACTLY ONE output number.

Be careful. Not every relation is a function. A function has to fit the above definition to a tee.

4

Functions

Chapter1

Domain

The domain is the set of all input values to which the rule applies. These are called your independent variables. These are the values that correspond

to the first components of the ordered pairs it is associated with.

Range

The range is the set of all output values. These are called your dependent variables. These are the values that correspond to the second components of the

ordered pairs it is associated with.

5

Functions

Chapter1

Function

x

x

x

x

y

y

y

y

y

Domain Range

Set A

Set B

6

Functions

Chapter1

Example (1): Determine whether the relation represents y as a function of x .

a) {(-2, 3), (0, 0), (2, 3), (4, -1)}

b) {(-1, 1), (-1, -1), (0, 3), (2, 4)}

Function

Not a Function

7

Functions

Chapter1

Definition of a Function

8

Functions

Chapter1

Input

x

The domain elements, x, can be thought of as the inputs and the range elements, f (x), can be thought of as the outputs.

Output

f (x)

Function

f

9

Functions

Chapter1

To evaluate a function f (x) at x = a, substitute the specified value a for x into the given function.

Example (1): Let f (x) = x2 – 3x – 1. Find f (–2).

f (x) = x2 – 3x – 1

f (–2) = 4 + 6 – 1 Simplify.

f (–2) = 9 The value of f at –2 is 9.

10

Functions

Chapter1

11

Functions

Chapter1

Example 3:  Find and simplify                             using the function                      .

12

Functions

Chapter1

Constant Function

A function of the form

f(x) = C ,

where C is a constant.Example 4: Find the functional values h)0( and h(2) of the constant function h(x)=-5

13

Functions

Chapter1

Compound Function

A compound function is also known as a piecewise function .

The rule for specifying it is given by more than one expression .

Example 5: Find the functional values f(1), f(3), and f(4) for the compound function

14

Functions

Chapter1

To find f(1), To find f(3), To find f(4),

15

Functions

Chapter1

Domain of a Function

16

Functions

Chapter1

17

Functions

Chapter1

18

Functions

Chapter1

x 2 + y

2 = 4

A relation is a correspondence that associates values of x with values of y.

y 2 = x

x

y

(4, 2)

(4, -2)

x

y

(0, 2)

(0, -2)

y = x 2

x

y

Example: The following equations define relations:

The graph of a relation is the set of ordered pairs (x, y) for which the relation holds.

19

Functions

Chapter1

y 2 = x

x

y

Vertical Line Test

A relation is a function if no vertical line intersects its graph in more than one point.

Of the relations y 2 = x, y = x 2, and x 2 + y 2 = 1 only y = x2 is a function. Consider the graphs.

x 2 + y

2 = 1

x

y

y = x 2

x

y

2 points of intersection

1 point of intersection

2 points of intersection

Functions

Vertical Line Test: Apply the vertical line test to determine which of the relations are functions.

The graph does not pass the vertical line test. It is not a function.

The graph passes the vertical line test. It is a function.

x

yx = 2y – 1

x

y x = | y – 2|

FunctionsDomain and Range

• In a relation, the set of all values of the independent variable (x) is the domain; the set of all values of the dependent variable (y) is the range.

Example• Give the domain and range of the

relation.

• The domain is {4, 5, 6, 8}; the range is {C, D, E}. The mapping defines a function—each x-value corresponds to exactly one y-value.

• Give the domain and range of the relation.

• The domain is {3, 4, 2} and the range is {6}. The table defines a function because each different x-value corresponds to exactly one y-value.

4

5

6

8

C

D

E62

64

63

yx

Functions

Finding Domains and Ranges from Graphs

Find the domain and range of the relation.

•The x-values of the points on the graph include all numbers

between 3 and 3, inclusive. The y-values include all numbers between 2 and 2, inclusive.

• Domain = [3, 3]• Range = [2, 2]

range

domain

Functions

Definitions

• Agreement on DomainUnless specified otherwise, the domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable.

• Vertical Line TestIf each vertical line intersects a graph in at most one point, then the graph is that of a function.

Functions

Example

• Use the vertical line test to determine whether each relation graphed is a function.

• The graph of (a) fails the vertical line test, it is not the graph of a function.• Graph (b) represents a function.

a. b.

Functions

Identifying Function, Domains, and Ranges from Equations

• Decide whether each relation defines a function and give the domain and range.

• a) y = x + 5

• b)

• c)

3 1y x

3

1y

x

Functions

Function Notation

• y = f(x) is called function notation

• We read f(x) as “f of x” (The parentheses do not indicate multiplication.) The letter f stand for function.

• f(x) is just another name for the dependent variable y.

• Example: We can write y = 10x + 3

as f(x) = 10x + 3

Functions

Solutions• a) y = x + 5

– Function: Each value of x corresponds to just one value of y so the relation defines a function.

– Domain: Any real number of (, )

– Range: Any real number (, )

Functions

Solutions continued

• b)

– Domain: [1/3, )

– Function: For any choice of x there is exactly one corresponding y-value. So the relation defines a function.

– Range: y 0, or [0, )

3 1y x

Range

Domain

13

3 1 0

3 1

x

x

x

Functions

Solutions continued• c)

– Function: There is exactly one value of y for each value in the domain, so this equation defines a function.

– Domain: All real numbers except those that make the denominator 0. (, 1) (1, ).

– Range: Values of y can be positive or negative, but never 0. The range is the interval (, 0) (0, ).

3

1y

x

Domain

Range

Functions

Example 8:

Solution to Example 8:

Functions

Functions

Functions

Functions

Functions

Functions

Variations of the Definition of Function

• A function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component.

• A function is a set of ordered pairs in which no first component is repeated.

• A function is a rule or correspondence that assigns exactly one range value to each domain value.

37

Chapter1

Examples on:

-Function Graph.

-Domain and Range of Functions

38

Functions

Chapter1

Example (1):

39

Functions

Chapter1

Example (2):

40

Functions

Chapter1

41

Functions

Chapter1

42

Functions

Chapter1

43

Functions

Chapter1

Example (3):

44

Functions

Chapter1

45

Functions

Chapter1

Example (4):

46

Functions

Chapter1

47

Functions

Chapter1

Example (5):

48

Functions

Chapter1