unit 1: functions lesson 1: relations and functions learning goals: i can determine if a relation is...

Unit 1: Functions

Lesson 1: Relations and Functions

Learning Goals:

I can determine if a relation is a function

Unit 1: Functions

Lesson 1: Relations and Functions

So far, we have seen mathematical relationships written like this:

•y = 3x + 1•y = 2x2 -2•y = x2

•y = 5x•Etc, etc.

These examples are relations: They are rules describing the relationship between the dependent and independent variables.

The Dependent Variable is:The Independent Variable is:

A relation is a connection (or relationship) between two sets of numbers, such as height vs. time or cost vs. weight

Unit 1: Functions

Lesson 1: Relations and Functions

Example: The height, h, of an object thrown up in the air is dependent on the time, t. “h” is dependent on “t”, therefore h is the dependent variable and t is the independent variable.

Unit 1: Functions

Lesson 1: Relations and Functions

These examples represent function notation and are read as, “ f of x”, or “f at x”.

Unit 1: Functions

Lesson 1: Relations and Functions

Function notation represents a relation where there is only one unique value of the function (f) for any value of x.In other words, each x-value (independent variable) has only one y-value (dependent variable)

Unit 1: Functions

Lesson 1: Relations and Functions

How do you know whether something is a function?

• If you put in a value for “x” and there is only one value for “y” it is a function.

• If you put in a value for “x” and get more than one value for “y”, it is not a function.

Example:

Camary

Rav 4

Yaris

Prius

Toyota

INPUT OUTPUT

CamaryVenzaRav 4SiennaYarisCorollaPrius

Toyota

INPUT OUTPUT

Unit 1: Functions

Lesson 1: Relations and Functions

Unit 1: Functions

Lesson 1: Relations and Functions

A function can be represented by:

1) A Table of Values2) A Set of Ordered Pairs3) A Mapping Diagram4) A Graph5) An Equation

Unit 1: Functions

Lesson 1: Relations and Functions

Table of Values: It is a function if each x-value only

corresponds to one y-value

x y-2 3-1 20 1-1 0

x y1 53 67 82 8

Unit 1: Functions

Lesson 1: Relations and Functions

Ordered Pairs: It is a function if for each x-

value there is only one y-value

f = {(1,-4), (2, 5), (8, 9), (0, 6)}

g = {(1, -3), (2, -3), (3, 0), (2, 0)}

Mapping Diagram: It is a function if the x-value points to only one y-value

1

0

-1

1

2

3

4

1

4

7

10

11

8

10

9

x y yx

Unit 1: Functions

Lesson 1: Relations and Functions

Unit 1: Functions

Lesson 1: Relations and Functions

Graph: It is a function if it passes the vertical line test. Vertical Line Test: Draw a vertical line through the graph. If the line crosses the graph more than once it is not a function.

Unit 1: Functions

Lesson 1: Relations and Functions

Equation: Anything in the form y = mx + b is a function. Anything in the form y = ax2+ bx + c is a function. To check anything else, graph it!

Unit 1: Functions

Lesson 1: Relations and Functions

a) y = 2x + 1 b) y = 2x2 - 3 c) x2 + y2 = 4

Determine whether the following relations are functions or not

Unit 1: Functions

Lesson 1: Relations and Functions

Domain: The set of all the input values that are defined for a function. (Formerly referred to as the x-values or the independent variable.) Written from smallest to largest number.Range: The set of all the output values for the function. Can be determined by subbing in the values from the domain. (Formerly referred to as the y-values or the dependent variable.) Also written from smallest to largest number.

Unit 1: Functions

Lesson 1: Relations and Functions

Unit 1: Functions

Lesson 1: Relations and Functions

Example: Write the domain and range for this function using set notation.

x y1 53 67 82 8

Unit 1: Functions

Lesson 1: Relations and Functions

Example: Write the domain and range for this function using set notation.

f = {(1,-4), (2,5), (8, 9), (0, 6)}

Unit 1: Functions

Lesson 1: Relations and Functions

Example: Write the domain and range for this function using set notation.

1

4

7

10

11

8

10

9

x y

Unit 1: Functions

Lesson 1: Relations and Functions

Example: Write the domain and range for this function using set notation.

Unit 1: Functions

Lesson 1: Relations and Functions

Practice

Level 4: pg. 10-12 # 1 – 12, 14

Level 3: pg. 10-12 # 1 – 10, 14

Level 2: Pg. 10-12 #1-7, 14

Level 1: Pg. 10-12 #1 -3, 14