2.1 Functions and their Graphs

Relations A relation is a mapping, or pairing, of input values with output values. The set of input values is called the domain. The set of output values is called the range. A relation is a function provided there is exactly one output for each input. It is NOT a function if at least one input has more than one output

Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 -2 4 1 4 Domain = {-3, 1,4} Range = {3,-2,1,4} Notice the set notation!!! Function? No: input 1 is mapped onto Both -2 & 1

Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 1 3 -2 4 Function? Yes: each input is mapped onto exactly one output Domain = {-3, 1,3,4} Range = {3,1,-2}

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

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

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

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

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 not 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

Use the vertical line test to visually check if the relation is a function. (4,4) (-3,3) (1,1) (1,-2) Function? No, Two points are on The same vertical line.

Use the vertical line test to visually check if the relation is a function. (-3,3) (1,1) (3,1) (4,-2) Function? Yes, no two points are on the same vertical line

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 Construct a table of values Graph enough solutions to recognize a pattern Connect the points with a line or curve

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

Function Notation f(x) = mx + b 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

Evaluating Functions Let f(x) = -x2 – 3x + 5 Find f(-2) = -(-2)2 – 3(-2) + 5 f(-2) = 7 Let g(x) = 2x + 6 g(-2) = 2(-2) + 6 g(-2) = 2

Evaluating Functions Let f(x) = -x2 – 3x + 5 Let g(x) = 2x + 6 Find f(2)·g(2) Find f(3) – g(1) Find f(4) + g(3) Find f(2)/g(1)

Assignment Complete Worksheet 1.1 on Functions and Relations