Section 1.2Basics of Functions

Definition of a Relation

A relation can be expressed as a set of ordered pairs. The domain of a relation is the set of first elements

in the ordered pairs, and the range is the set of second elements.

Relation: {(0 , 5), (2 , 1), (2 , 1), (3 , 8)}

Math 112 Section 1.2

Relation: {(0 , 5), (2 , 1), (2 , 1), (3 , 8)}

Domain: {0, 2, 3}

Relation: {(0 , 5), (2 , 1), (2 , 1), (3 , 8)}

Domain: {0, 2, 3}

Range: {5, 1, 1, 8}

Example

Find the domain and the range.

98.6, Felicia , 98.3,Gabriella , 99.1, Lakeshia

Definition of a Function

A function is a relation for which each element of the domain corresponds to exactly one element of the range.

Relation: {(0 , 5), (2 , 1), (2 , 1), (3 , 8)}Function: {(0 , 2), (1 , 8), (5 , 2), (1 , 3)}

In other words, no x coordinate can be paired with more than one y coordinate.

Math 112 Section 1.2

0 52 13 1

8

0 2 1 8 5 3 1

Example

Determine whether each relation is a function?

1,8 , 2,9 , 3,10

2,3 , 2,4 , 2,5

3,6 , 4,6 , 5,6

Function Notation

Function Notation

A function can also be expressed as an equation.

f(x) = x2 + 5x 2

f(3) = 32 + 5(3) 2 = 22

f(1) = (1)2 + 5(1) 2 = 6f(z+2) = (z+2)2 + 5(z+2) 2

= z2 + 4z + 4 + 5z + 10 2= z2 + 9z + 12

Math 112 Section 1.2

“f of x”

Example

Evaluate each of the following.

2

2

Find f(3) for f(x)=2x 4

Find f(-2) for f(x)=9-x

Example

Evaluate each of the following.

2

2

Find f(x+2) for f(x)=x 2 4 ?

Is this is same as f(x) + f(2) for f(x)=x 2 4

x

x

Example

Evaluate each of the following.

2

2

Find f(-x) for f(x)=x 2 4

Is this is same as -f(x) for f(x)=x 2 4?

x

x

Graphs of Functions

The graph of a function is the graph of its ordered pairs.

First find the ordered pairs, then graph the functions.

Graph the functions f(x)=-2x; g(x)=-2x+3

x f(x)=-2x (x,y) g(x)=-2x+3 (x,y)

-2 f(-2)=4 (-2,4) g(-2)=7 (-2,7)

-1 f(-1)=2 (-1,2) g(-1)=5 (-1,5)

0 f(0)=0 (0,0) g(0)=3 (0,3)

1 f(1)=-2 (1,-2) g(1)=1 (1,1)

2 f(2)=-4 (2,-4) g(2)=-1 (2,-1)

See the next slide.

x

y

x

y

f(x)g(x)

Example

Graph the following functions f(x)=3x-1 and g(x)=3x

x

y

The Vertical Line Test

x

y

The first graph is a function, the second is not.

x

y

x

y

Example

Use the vertical line test to identify graphs in which y is a function of x.

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y

Example

Use the vertical line test to identify graphs in which y is a function of x.

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Obtaining Information

from Graphs

Example

Analyze the graph.2( ) 3 4

a. Is this a function?

b. Find f(4)

c. Find f(1)

d. For what value of x is f(x)=-4

f x x x

x

y

Identifying Domain and Range from a Function’s Graph

The domain of a function is the set of all x values for which the function is defined.

Domain

x2 4 0

x 2, 2( , 2) (2 , 2) (2 , )

4x

2xf(x)

2

62x f(x)

Domain

2x + 6 0

2x 6x 3[3 , )

Math 112 Section 1.2

Finding the Domain & Range of a Function

The domain of a function is the set of all x values from the graph.

The range of a function is the set of all y values from the graph.

Domain: ( , )

Range: [1 , )

Math 112 Section 1.2

x

yIdentify the function's domain and range from the graph

Domain (-1,4]

Range [1,3)

Domain [3, )

Range [0, )

x

y

Example

Identify the Domain and Range from the graph.

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y

Example

Identify the Domain and Range from the graph.

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y

Example

Identify the Domain and Range from the graph.

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y

Identifying Intercepts

from a Function’s Graph

Example

Find the x intercept(s). Find f(-4)

x

y

Example

Find the y intercept. Find f(2)

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y

x

y

Example

Find the x and y intercepts. Find f(5).

Find f(7).

x

y

Find the Domain and Range.

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y

22 3( ) Find f(-1)

7

xf x

Example

Determine whether each equation defines y as a function of x.

2

2 2

4 8

2 10

16

x y

x y

x y