2.1 Functions and Their Graphs

2.1 Functions and Their Graphs

What is a Relation?•A relation is a mapping, or pairing, of input

values with output values.

•The set of input values is the domain.•The set of output values is the range.

•What are the domain and range of this relation?

“Mapping Diagram”

How Can We Write a Relation?•A relation can be written in the form

of a table:

•A relation can also be written as a set of ordered pairs:

How Do We Write a Relation with Numbers?

•Set of ordered pairs with form (x, y).•The x-coordinate is the input and the y-

coordinate is the output.•Example:

{ (0, 1) , (5, 2) , (-3, 9) }•{ } is the symbol for a “set”

•What is the domain and range of this relation?

How Do We Graph a Relation?•To graph a relation, plot each of its

ordered pairs on a coordinate plane.•Graph the relation: { (0, 1) , (5, 2) , (-3, 9) } Remember:

The x comes first – moves right or left. The y comes second – moves up or down.

Positive means to the right or up.Negative means to the left or down.

Your Turn!

•Graph the relation and identify the domain and range.{ (-1,2), (2, 5), (1, 3), (8, 2) }

What is a Function?

•A function is a special type of relation that has exactly one output for each input.

• If any input maps to more than one output, then it is not a function.

• Is this a function? Why or why not?

Which of These Relations Are Functions?

•

•{ (3,4), (4,5), (6,7), (3,9) }

• X 5 7 9 2 6

y 1 6 2 8 4

Using the Vertical Line Test•A relation is a function if and only if no

vertical line crosses the graph at more than one point.

•This is not a function because the vertical line crosses two points.

Your Turn!

•Write the domain and range. • Is this a function?

{ (2,4) (3,6) (4,4) (5, 10) }

stop

What is a Solution of an Equation?•Many functions can be written as an

equation, such as y = 2x – 7.•A solution of an equation is an ordered

pair (x, y) that makes the equation true.

•Example: Is (2, -3) a solution of y = 2x – 7 ?

What are Independent and Dependent Variables?

•The input is called the independent variable.▫Usually the x

•The output is called the dependent variable.▫Usually the y

•Helpful Hints: ▫Input and Independent both start with “in”▫The Dependent variable depends on the

value of the input

What Does the Graph of an Equation Mean?•The graph of a two variable equation is the

collection of all of its solutions.•Each point on the graph is an ordered pair

(x, y) that makes the equation true.

•Example: This is the graph of the equation y = x + 2

How Do We Graph Equations?•Step 1: Construct a table of values.

•Step 2: Graph enough solutions to notice a pattern.

•Step 3: Connect the points with a line or curve.

Example:•Graph the equation y = x + 1

Your Turn!

•Graph the equation y = x – 2

What is Function Notation?

•Function notation is another way to write an equation.

•We can name the function “f” and replace the y with f(x).

• f(x) is read “f of x” and means “the value of f at x.”▫Be Careful! It does not mean “f times x”

•Not always named “f”, they sometimes use other letters like g or h.

What is a Linear Function?

•A linear function is any function that can be written in the form f(x) = mx + b

• Its graph will always be a straight line.

•Are these functions linear?▫f(x) = x2 + 3x + 5▫g(x) = 2x + 6

How Do We Evaluate Functions?•Plug-in the given value for x and find f(x).

•Example: Evaluate the functions when x = -2.▫f(x) = x2 + 3x + 5

▫g(x) = 2x + 6

Your Turn!

•Decide if the function is linear. Then evaluate the function when x = 3.g(x) = -3x + 4

Stop?

How Do We Find the Domain and Range?•The domain is all of the input values that

make sense.▫Sometimes “all real numbers”▫For real-life problems may be limited

•The range is the set of all outputs.

Example:• In Oak Park, houses will be from 1450 to

2100 square feet. The cost C of building is $75 per square foot and can be modeled by C = 75f, where f is the number of square feet. Give the domain and range of C(f).