Intro to FunctionsNovember 30, 2015

Function? What’s That?A function is a relationship between input and output values where each input has exactly one outputRemember:Inputs are the “x-values”Outputs are the “y-values”

In other words:

Each x-value, can only have one y-value

What does it look like?There are multiple ways to represent a function.

A function can be represented as a mapping, table

of values, graph, or a rule. Mapping Table of ValuesIn Out

12 4

6 9

Graph Rule The in times four plus six equals the out.

Mappings use arrows to connect input and output values.

All inputs (x-values) are listed on the left

All outputs (y-values) are listed on the right

In (X) Out (Y)4691213

035811Arrows show the output

(y-value) for each input (x-value)

How do I know if a mapping is a function?

Recall: In a function, each input has exactly one output.

If each input connects to one output in a mapping, then the mapping is a function.

In (X) Out (Y)4691213

035811

Write the ordered pairs given in the Mapping

In (X) Out (Y)4691213

035811

(4,3)(6,5)(9, 11)(12, 0)(13, 8)

Is the mapping a function?

Yes!!! YES!!! NO!!!

But why?

Each input connects to one output

Each input connects to one outputNOTE: In a function, two inputs can have the same output

-8 connects to both -6 and 1. Inputs can only connect to one output in a function

1)

2)

3)

Table of Values lines-up input and output values with inputs listed least to greatest

How do I know if a table of values is a function?Recall: In a function,

each input has exactly one output. If each input

appears only once in the table, the table represents a function

Is the table of values a function?In

Out

-2 2-1 40 61 9

In Out

4 75 145 -610 19

In Out

3 05 417 629 0

1)

2)

3)

YES!!! NO!!! YES!!!

But why?

Each input value appears once (Each input has one output)

Each input value appears once (Each input has one output)In a function, two inputs could have the same output

The input value 5 appears more than once (it has two different output values)

Intro to Functions Day 212/1/15

Representing Functions with Rules and Function Notation

Representing a ruleWhen writing a rule, we have been using complete sentences.

From this point, we will be using mathematical notation to represent the rules.

What do we call a mathematical sentence? EQUATIO

N

Writing the rule with mathematical notation

In Out-1 00 -52 -15

What is the rule using a complete sentence?

The in times -5 minus 5 equals the out.

As an equation?

−5 𝑥−5=𝑦

Function Notation

Equations that are functions are written in a form called “function notation”

Because a function is a special type of equation (the input has one output), we name it in a special way.

Examples of Function Notation:• f(x) Reads as “f of x” Means “The function named f

with input x• g(x) Reads as “g of x” Means “The function named g with input x• h(x) Reads as “h of x” Means “The function named h with input x

Writing a rule in function notation

−5 𝑥−5=𝑦

𝑓 (𝑥 )=−5 𝑥−5

Is this Table of Values a Function?

In Out-1 00 -52 -15

Rule as an Equation

YES!Rule in function

notation

Represent the Function as a Rule

𝑓 (𝑥 )=𝑥+5 h (𝑥 )=𝑥÷3 g (𝑥 )=2𝑥+1

Big Idea HereIf it’s a function, its rule is named with function notation Example:

f(x) = 6x + 7g(x) = 7x - 3h(x) =

Knowledge Check Problem1) Represent the mapping as a table of values.

2) Is it a function?3) If it is a function, find the rule as write it using function notation.

IN OUT-201367

113-31537

Intro to Functions Cont.

12.7.2015

Focus: Graphs of Functions and Intro to Linear Functions

Graphs of Functions Ordered Pairs: (2, -7) (2, -5) (2, 0) (2, 3) (2, 4)Just looking at the ordered pairs, would these

ordered pairs represent a function?

Why?

No!

The input 2 has 5 different output!

Plot the ordered pairs and connect the points. (2, -7) (2, -5) (2, 0) (2, 3) (2, 4)What do you notice about the graph you just made?The points line up

vertically

Let’s look at another setOrdered pairs( 5, -2) (1, -1) (-2, 1) (0, 2) (3, 3) ( 5, 4)Based on just the ordered pairs, would this be a function?

No

Graph the points and connect them in order

Why?The input 5 has 2 outputs

In this graph, we said it’s not a function because the input 2 has 5 different outputs.All those points lines up vertically.

In this graph, we said it’s not a function because the input 5 has 2 different outputs.

What similarities do you see?

Vertical Line Test

If a graph passes the vertical line test, then the graph is a function.

When given a graph, draw vertical lines throughout the graph. If all of the lines you draw cross the graph only once then the graph passes the vertical line test. If AT LEAST one line you draw crosses the graph more than once, then the graph fails the vertical line test.

Try it out:

Is it a function? Is it a function?YES!!!

NO!!!

Graphs of Functions Important Features Domain and Range

• The domain is all of the input values (listed least to greatest)• The range is all of the output values

(listed least to greatest)A graph of a function includes the points (-2, 3) (4, 9) (-5, 6) (7, 12) (-3, -34) ( 0, -3)

Domain: { -5, -3, -2, 0, 4, 7}Range: { -34, -3, 3, 6, 9, 12}

Write the domain and range1. (4, 5) ( -3, 2) (0, -9) (12, -24) (-4, 5) ( -5, 3)

2. (-2, 3) ( 12, -11) (-6, 8) (-4, 5) ( 10, 3) ( 22, -34) D: {-5, -4, -3, 0,

4, 12}R: {-24, -9, 3, 2, 5}*Notice “5” is only

written in the range once

D: {-6, -4, -2, 10, 12, 22}R: {-34, -11, 3, 5, 8}*“3” is only written

in the range once

Linear Function:A function that is a straight line.

Linear Function: Key Parts1.y-intercept

The point where the line crosses the y-axis. 2. SlopeHow the graph changes

(0,1)

Slope of a lineIf we are given any two points (x1, y1) and (x2, y2) on a line we can calculate the slope of the line as follows:

x

y

x2 – x1

(x1, y1)

(x2, y2)

y2 – y1Draw a right-angled triangle between the two points on the line as follows:

𝒔𝒍𝒐𝒑𝒆=𝒓𝒊𝒔𝒆𝒓𝒖𝒏 𝒐𝒓 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒚

𝒄𝒉𝒂𝒏𝒈𝒆𝒊𝒏𝒙

𝒔𝒍𝒐𝒑𝒆=𝒚𝟐−𝒚𝟏

𝒙𝟐−𝒙𝟏

Slope of a line1. Find two points on the line

(1, 2)

(5, 5)

(1, 2) => ((5, 5) => (

2. Set it up

𝟓−𝟐𝟓−𝟏=

𝟑𝟒

This means, the graph moves up three points, and to the right 4 points.

𝒔𝒍𝒐𝒑𝒆=𝒚𝟐−𝒚𝟏

𝒙𝟐−𝒙𝟏

𝒚𝟐−𝒚 𝟏

𝒙𝟐−𝒙𝟏

Try it 1. What’s the y-intercept?

2. What’s the slope?

(0, 3)

1. Points to use: (-2, 0) and (2, 6)

𝟔−𝟎𝟐−(−𝟐)

=¿𝟔𝟒 𝒔𝒊𝒎𝒑𝒍𝒊𝒇𝒚⇒ 𝟑𝟐

So the graph moves up three points and to the right 2 points

(-2, 0)

(2, 6)