Intro to Functions November 30, 2015

A function is a relationship between input and output values where each input has exactly one output Remember: Inputs are the “x-values” Outputs are the “y-values” In other words: Each x-value, can only have one y-value

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

All inputs (x-values) are listed on the left All outputs (y-values) are listed on the right In (X)Out (Y) Arrows show the output (y- value) for each input (x-value)

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)

In (X)Out (Y) (4,3) (6,5) (9, 11)(12, 0) (13, 8)

Yes!!! YES!!!NO!!! But why? Each input connects to one output NOTE: 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

In Out In Out In Out ) 2)3) YES!!! NO!!! YES!!! But why? 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 2 12/1/15 Representing Functions with Rules and Function Notation

When 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? EQUATION

InOut What is the rule using a complete sentence? The in times -5 minus 5 equals the out. As an equation?

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

Is this Table of Values a Function? InOut Rule as an Equation YES! Rule in function notation

If it’s a function, its rule is named with function notation

1) 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

Focus: Graphs of Functions and Intro to Linear 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

Ordered 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?

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.

Is it a function? YES!!! NO!!!

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}

1. (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.

1.y-intercept The point where the line crosses the y-axis. 2. Slope (0,1)

If we are given any two points ( x 1, y 1 ) and ( x 2, y 2 ) on a line we can calculate the slope of the line as follows: x y x 2 – x 1 ( x 1, y 1 ) ( x 2, y 2 ) y 2 – y 1 Draw a right-angled triangle between the two points on the line as follows:

1. Find two points on the line (1, 2) (5, 5) 2. Set it up This means, the graph moves up three points, and to the right 4 points.

1. What’s the y-intercept? 2. What’s the slope? (0, 3) 1. Points to use: (-2, 0) and (2, 6) So the graph moves up three points and to the right 2 points (-2, 0) (2, 6)