2.1 Functions and their Graphs page 67

Learning Targets I can determine whether a given relations is a function. I can represent relations and function. I can graph and evaluate linear functions.

Relations A relation is a mapping, or pairing, of input values with output values. The set of input values is called the domain. Also called x-coordinate. The set of output values is called the range. Also called y-coordinate. A relation as a function provided there is exactly one output for each input. NOTE: x values do not repeat. It is NOT a function if at least one input has more than one output

Functions A function is a relation in which the members of the domain (x- values) DO NOT repeat. So, for every x-value there is only one y-value that corresponds to it. y-values can be repeated.

Input (x-values) Output (y-values) 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. X repeats. Notice the set notation!!!

Identify the Domain and Range. Then tell if the relation is a function. Input Output Domain = {-3, 1,3,4} Range = {3,1,-2} Function? Yes: each input is mapped onto exactly one output x values do not repeat

A Relation can be represented by a set of ordered pairs of the form (x,y) Quadrant I X>0, y>0 Quadrant II X 0 Quadrant III X<0, y<0 Quadrant IV X>0, y<0 Origin (0,0)

Graphing Relations To graph the relation in the previous example: Write as ordered pairs (-3,3), (1,-2), (1,1), (4,4) Plot the points

(-3,3) (4,4) (1,1) (1,-2)

Same with the points (-3,3), (1,1), (3,1), (4,-2)

(-3,3) (4,-2) (1,1) (3,1)

Vertical Line Test You can use the vertical line test to visually determine if a relation is a function. Slide any vertical line (pencil) across the graph to see if any two points lie on the same vertical line. If there are no 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) (1,1) (1,-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

x y x y Does the graph represent a function? Yes

x y x y Does the graph represent a function? No

Does the graph represent a function? Yes No x y x y

Graphing and Evaluating Functions Many functions can be represented by an equation in 2 variables: y=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=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 y=2x-7 ….. X is the independent var. Y is the dependant var. 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 variables 1.Construct a table of values 2.Graph enough solutions to recognize a pattern 3.Connect the points with a line or curve

Graph: y = x + 1 Step 1 Table of values Step2: Step 3:

Function Notation By naming the function ‘f’ you can write the function notation: f(x) = mx + b “the value of f at x” “f of x” f(x) is another name for y (grown up name) You can use other letters for f, like g or h

Decide if the function is linear. Then evaluate for x = -2 f(x) = -x 2 – 3x + 5 Not linear…. f(-2) = -(-2) 2 – 3(-2) + 5 f(-2) = 7 g(x) = 2x + 6 Is linear because x is to the first power g(-2) = 2(-2) + 6 g(-2) = 2 The domain for both is….. All reals

Pair-Share Pp #5-48 (Even Number Only)