Chapter 2: Functions and Graphs

Chapter 2: Functions and Graphs

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What the correlation ?What the correlation ?

There are other ways to describe relations between variables.

Set to set

Ordered pairs

• A set of ordered pairs (x, y) is also called a relation.

• The domain is the set of x-coordinates of the ordered pairs.

• The range is the set of y-coordinates of the ordered pairs.

Find the domain and range of the relation

{(4,9), (-4,9), (2,3), (10,-5)}

• Domain is the set of all x-values, {4, -4, 2, 10}.

• Range is the set of all y-values, {9, 3, -5}.

Example 1

Example 2

Domain:{Polar Bear, Cow, Chimpanzee, Giraffe, Gorilla, Kangaroo, Red Fox}

Range:

{20, 15, 10, 7}

• Some relations are also functions.

• A function is a set of order pairs in which each first component in the ordered pairs corresponds to exactly one second component.

x

DOMAIN

y

RANGE

f

FUNCTION CONCEPT

x

DOMAIN

y1

y2

RANGE

R

NOT A FUNCTION

y

RANGE

f

FUNCTION CONCEPT

x1

DOMAIN

x2

Ways to Represent a FunctionWays to Represent a Function

• SymbolicSymbolic

x,y y 2x or

y 2x

X Y

1 2

5 10

-1 -2

3 6

• GraphicalGraphical

• NumericNumeric

• VerbalVerbalThe cost is twice the original amount.

y f x

• Output Value• Member of the Range• Dependent Variable

These are all equivalent names for the y.

• Input Value• Member of the Domain• Independent Variable

These are all equivalent names for the x.

Name of the function

FUNCTION NOTATIONFUNCTION NOTATION

Example 3

y = -3x + 2 so represents a function.

• We often use letters such as f, g, and h to name functions.

• We can use the function notation f(x) (read “f of x”) and write the equation as f(x) = -3x + 2.

Note: The symbol f(x) is a specialized notation that does NOT mean f • x (f times x).

• to evaluate a function at x substitute the x-value into the notation.

• Example 4

f(x) = -3x + 2, f(2) = -3(2) + 2 = -6 + 2 = -4.

Example 5

g(x) = x2 – 2x

a.find g(-3)

b.write down the corresponding ordered pair.

Answer :

• g(-3) = (-3)2 – 2(-3) = 9 – (-6) = 15.

• The ordered pair is (-3, 15).

Drawing Graphs of Functions

A way to visualize a function is by drawing its graphThe graph of a real function f of one variable is the set of all points P(x, y) in the plane such that y = f(x). Plot the value of x on the horizontal, or x-axis and the value of f(x) on the vertical, or y-axis. How can we tell whether a set of points in the plane is the graph of some function? By reading the definition of a function again, we have an answer.

Graphs of functions??

Ex 6.

Given the relation {(4,9), (-4,9), (2,3), (10,-5)}, is it a function?

•Since each element of the domain (x-values) is paired with only one element of the range (y-values) , it is a function.

Note:

Each x-value has to be assigned to ONLY

one y-value!!!

Domain and Range

Is the relation y = x2 – 2x a function?

• Since each element of the domain (the x-values) would produce only one element of the range (the y-values), it is a function.

Question:What does the graph of this function look like?

Does this graph pass

the vertical line test?

Example 7

8

6

4

2

-2

-4

-6

-10 -5 5 10

f x = x2-x

Is the relation x2 + y2 = 9 a function?

• Since each element of the domain (the x-values) would correspond with 2 different values of the range (both a positive and negative y-value), the relation is NOT a function

Check the ordered pairs: (0, 3) (0, -3)

The x-value 0 corresponds to two different y-values, so the relation is NOT a function.

Question: What does the graph of this relation look like?

Example 8

Ex

Domain

Range

Example 9

Use the vertical line test to determine whether the graph to the right is the graph of a function.

x

y

Since no vertical line will intersect this graph more than once, it is the graph of a function.

Example 10

Use the vertical line test to determine whether the graph to the right is the graph of a function.

x

y

Since no vertical line will intersect this graph more than once, it is the graph of a function.

Example 11

Use the vertical line test to determine whether the graph to the right is the graph of a function.

Since a vertical line can be drawn that intersects the graph at every point, it is NOT the graph of a function.

x

y

Example 12

Use the vertical line test to determine whether the graph to the right is the graph of a function.

Since vertical lines can be drawn that intersect the graph in two points, it is NOT the graph of a function.

x

y

Find the domain and range of the function graphed (in red) to the right. Use interval notation.

x

y

Domain is [-3, 4]

Domain

Range is [-4, 2]

Range

Determining the domain and range from the graph of a relation:

Example:

Example 13

Find the domain and range of the function graphed to the right. Use interval notation. x

y

Domain is (-, ) DomainRange is [-2, )

Range

Example 14

Find the domain and range of the function graphed to the right. Use interval notation.

x

y

Domain: (-, )

Range: (-, )

Example 15

Find the domain and range of the function graphed to the right. Use interval notation.

x

y

Domain: (-, )Range: [-2.5](The range in this case consists of one single y-value.)

Example 16

Find the domain and range of the relation graphed to the right. Use interval notation.

(Note this relation is NOT a function, but it still has a domain and range.)

Domain: [-4, 4]

Range: [-4.3, 0]

x

y

Example 17

Find the domain and range of the relation graphed to the right. Use interval notation.(Note this relation is NOT a function, but it still has a domain and range.)

Domain: [2]

Range: (-, )

x

y

Decide the Domain and Range: Graph

Homework11

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Decide the Domain and Range: Graph

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2:

yRyRf

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