Functions and Functions and their Graphstheir Graphs

RelationsRelations• A relation is a mapping of input values with

output values.

• The set of x-values (input values) is called the domain.

• The set of y-values (output values) is called the range.

• A relation is a function provided there is exactly one output for each input. Each element of the domain is paired with only one element of the range

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

Input Output

-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

Notice the set notation!!!

Identify the Domain and Range. Then Identify the Domain and Range. Then

tell if the relation is a function.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 output

Vertical Line TestVertical Line Test

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

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

• If there are not 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)

(2,2)

(2,-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

Graphing and Evaluating Graphing and Evaluating

FunctionsFunctions

• Many functions can be represented by an equation in 2 variables: y = 2x - 7 or! f(x) = 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 = f(x) = 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 f(x) = 2x-7 ….. x is the independent variable and y is the dependent variable

• 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 Graphing an equation in 2

variablesvariables

1.Construct a table of values

2.Graph enough solutions to recognize a pattern

3.Connect the points with a line or curve

Graph: f(x)=y = Graph: f(x)=y = x + 1x + 1

Step 1Table of values

Step2:Step 3:

PracticePracticeCreate a table with 5 different values. Graph the lines on the coordinate plane.

1.f(x) = 2x+ 3

2.h(x) = - 3x+1

3.g(x)= 5 – x

2.R(x)= x - 4

More Practice!More Practice!1. Given f(x) = 9x - 1, find f(0).

2. If h(x) = -3, find x in h(x) = 7x + 4

3. Mary is machine saleswoman who earns a base salary of $3,000 plus a commission $200 for every machine she sells. Write a functions (equation) that shows the total income Mary earns if she sells x machines in one month. How much money will Mary make in April if she sells 11 machines?

4. Paul opens a savings account with $350 dollars. He saves $150 per month. Assume that he does not withdraw money or

make any additional deposits. a). Write a linear model that represents the total amount of money Paul has in his account after m months.

b). After how many months will Paul have more than $2,000?

1. f(0))=9(0) – 1 = - 1 2. If h(x) = -3, in h(x) = 7x + 4, then -3 = 7x + 4 Solve for x and x = -1 3. Mary’s salary: y = 3,000 + 200x (x the number

of machines she sells, y her monthly salary) If Mary sells x = 11 machines, then y = 3,000 + 200(11) = $5,2004. a) y = 350 + 150m (m = month, y = money in savings account)b) 2000 = 350 + 150m and solve for m; m = 11 months