2.1 Functions and their Graphs

page 67

Learning Targets

• I can determine whether a given relations is a function.

• I can represent relations and function.

• I can graph and evaluate linear functions.

Relations• A relation is a mapping, or pairing, of

input values with output values.• The set of input values is called the

domain. Also called x-coordinate.• The set of output values is called the

range. Also called y-coordinate.• A relation as a function provided

there is exactly one output for each input. NOTE: x values do not repeat.

• It is NOT a function if at least one input has more than one output

Functions

• A function is a relation in which the members of the domain

(x-values) DO NOT repeat.

• So, for every x-value there is only one y-value that corresponds

to it.• y-values can be repeated.

Input (x-values) Output (y-values)

-3 3

1 -2

4 1

4

Identify the Domain and Range. Then tell if the relation is a function.

Domain = {-3, 1,4}Range = {3,-2,1,4}

Function?No: input 1 is mapped onto Both -2 & 1 . X repeats.

Notice the set notation!!!

Identify the Domain and Range. Then tell if the relation is a function.

Input Output

-3 3

1 1

3 -2

4

Domain = {-3, 1,3,4}Range = {3,1,-2}

Function?Yes: each input is mappedonto exactly one outputx values do not repeat

A Relation can be represented by a set of ordered pairs of the form (x,y)

Quadrant IX>0, y>0

Quadrant IIX<0, y>0

Quadrant IIIX<0, y<0

Quadrant IVX>0, y<0

Origin (0,0)

Graphing Relations

• To graph the relation in the previous example:

• Write as ordered pairs (-3,3), (1,-2), (1,1), (4,4)

• Plot the points

(-3,3)(4,4)

(1,1)

(1,-2)

Same with the points (-3,3), (1,1), (3,1), (4,-2)

(-3,3)

(4,-2)

(1,1) (3,1)

Vertical Line Test

• You can use the vertical line test to visually determine if a relation is a function.

• Slide any vertical line (pencil) across the graph to see if any two points lie on the same vertical line.

• If there are no two points on the same vertical line then the relation is a function.

• If there are two points on the same vertical line then the relation is NOT a function

(-3,3)(4,4)

(1,1)

(1,-2)

Use the vertical line test to visually check if the relation is a function.

Function?No, Two points are on The same vertical line.

(-3,3)

(4,-2)

(1,1) (3,1)

Use the vertical line test to visually check if the relation is a function.

Function?Yes, no two points are on the same vertical line

x

y

x

y

Does the graph represent a function?

Yes

Yes

x

y

x

y

Does the graph represent a function?

No

No

Does the graph represent a function?

Yes

Nox

y

x

y

Graphing and Evaluating Functions

• Many functions can be represented by an equation in 2 variables: y=2x-7

• An ordered pair is a solution if the equation is true when the values of x & y are substituted into the equation.

• Ex: (2,-3) is a solution of y=2x-7 because:• -3 = 2(2) – 7• -3 = 4 – 7• -3 = -3

• In an equation, the input variable is called the independent variable.

• The output variable is called the dependent variable and depends on the value of the input variable.

• In y=2x-7 ….. X is the independent var. Y is the dependant var.

• The graph of an equation in 2 variables is the collection of all points (x,y) whose coordinates are solutions of the equation.

Graphing an equation in 2 variables

1. Construct a table of values

2. Graph enough solutions to recognize a pattern

3. Connect the points with a line or curve

Graph: y = x + 1

Step 1Table of values

Step2:Step 3:

Function Notation • By naming the function ‘f’ you can write

the function notation:

• f(x) = mx + b• “the value of f at x”

• “f of x”

• f(x) is another name for y (grown up name)

• You can use other letters for f, like g or h

Decide if the function is linear. Then evaluate for x = -2

• f(x) = -x2 – 3x + 5• Not linear….• f(-2) = -(-2)2 – 3(-2) + 5• f(-2) = 7

• g(x) = 2x + 6• Is linear because x is to the first power• g(-2) = 2(-2) + 6• g(-2) = 2• The domain for both is…..• All reals

Pair-Share

• Pp. 71-72 #5-48

(Even Number Only)