2.1 Functions and Their Graphs

In this lesson you will: Represent relations and functions. Graph and evaluate linear functions.

Warm Up

Recall the Cartesian Coordinate Plane y-axis 10 Quadrant II 8 Quadrant I 6 4 Origin 2 -10 -8 -6 -4 -2 O 2 4 6 8 10 x-axis -2 -4 Quadrant III Quadrant IV y-coordinate -6 -8 (10,-8) x-coordinate

A RELATION is a mapping or pairing of input values with output values A RELATION is a mapping or pairing of input values with output values. Here are ways to represent this pairing. Domain (x-input) 2 -1 5 7 Range (y-output) 7 8 10 MAPPING (-1,8), (2,8), (5,7), (5,10), (7,10) SET OF ORDERED PAIRS: 4 2 6 -2 -4 -6 8 10 -8 -10 x y GRAPH TABLE x y -1 8 2 5 7 10

FUNCTION A function is a special type of relation in which each element of the domain is paired with exactly one element from the range. Domain 3 2 5 8 Range 6 7 9 Range 8 Domain 3 1 4 8

What is the domain and range of the relation shown in the graph What is the domain and range of the relation shown in the graph. Is a relation or a function? 4 2 6 -2 -4 -6 8 10 -8 -10 x y Domain -4 -6 2 Range -4 2 6 Since one element of the domain is paired with two elements from the range. This is not a function. Domain: {-6, -4, 2} Range: {-4, 2, 4, 6}

Express the relation below as a mapping, a table and a graph, and state the domain and range. Is it a function? TABLE (-2,3), (4,1), (5,5), (7,-1), (7,8) x y -2 3 4 1 5 7 -1 8 MAPPING x 4 -2 5 7 y 1 -1 3 5 8 One element in the domain is paired with two of the range. This is not a function 4 2 6 -2 -4 -6 8 10 -8 -10 x y GRAPH A vertical line crossing two points also shows that this is not a function! D={-2, 4, 5, 7} R={-1, 1, 3, 5,8}

Determine if the graphs below are functions? 4 2 6 -2 -4 -6 8 10 -8 -10 x y 4 2 6 -2 -4 -6 8 10 -8 -10 x y NOTE: Drawing a vertical line we see it may cross the graph in only one point so IT IS A FUNCION! NOTE: Drawing a vertical line we see it crosses the graph in TWO POINTS so it is NOT A FUNCTION

Determine if the graphs below are functions? 4 2 6 -2 -4 -6 8 10 -8 -10 x y 4 2 6 -2 -4 -6 8 10 -8 -10 x y Drawing a vertical line we see it may cross the graph in only one point so IT IS A FUNCION! Drawing a vertical line we see it crosses the graph in TWO POINTS so it is NOT A FUNCTION

Determine if the graphs below are functions? 4 2 6 -2 -4 -6 8 10 -8 -10 x y 4 2 6 -2 -4 -6 8 10 -8 -10 x y Drawing a vertical line we see it may cross the graph in only one point so IT IS A FUNCION! Drawing a vertical line we see it crosses the graph in TWO POINTS so it is NOT A FUNCTION

Determine if the graphs below are functions? 4 2 6 -2 -4 -6 8 10 -8 -10 x y 4 2 6 -2 -4 -6 8 10 -8 -10 x y Drawing a vertical line we see it may cross the graph in only one point so IT IS A FUNCION! Drawing a vertical line we see it crosses the graph in TWO POINTS so it is NOT A FUNCTION We will call this the VERTICAL LINE test. It will help us decide if a relation is a FUNCTION.

Function Notation By naming a function “f”, you can write the function using function notation: The symbol f(x) is read a “the value of f at x” or simply “f of x”. Note that f(x) is another name for y. The input values for x are the domain of f. The output values for f(x) or y are the range of f. Function names could be represented by other names than f, such as g(x), or even names like abs(x) or sin(x).

f(1)=2(1)+1=3 f(-3)=2(-3)+1=-5 f(k)=2(k)+1=2k+1 If f(x) = 2x +1 f(1)=2(1)+1=3 f(-3)=2(-3)+1=-5 f(k)=2(k)+1=2k+1 f(x+h)=2(x+h)+1=2x+2h+1 f( ) = 2( ) + 1

Evaluate the following functions with the given values: g(x) = 2x + 1 and h(x)= 4x -1 2 2 4( ) -1 g(3) = 2( ) + 1 3 h(2) = h(0) = h(z) = 2 = 4(4)-1 = 6 + 1 = 7 = 16 -1 =15 g(-5) = 2( ) + 1 -5 = -10 + 1 2 4( ) -1 = -9 = 0 -1 =-1 g(k) = 2( ) + 1 k = 2k + 1 2 4( ) -1 z g(r+1) = 2( ) + 1 = 4z -1 2 r+1 = 2r + 2 + 1 = 2r + 3

x y (x,y) -1 1 2 2( ) -2( )- 1 2( ) -2( )- 1 2( ) -2( )- 1 Evaluate the following equation for the given domain and then determine if it represents a function. y = 2x - 2x - 1 2 D={-1, 0, 1, 2} 4 2 6 -2 -4 -6 8 10 -8 -10 x y x y (x,y) -1 1 2 2x - 2x - 1 2 2 2( ) -2( )- 1 3 -1 (-1,3) 2 2( ) -2( )- 1 (0,-1) -1 2 2( ) -2( )- 1 1 -1 (1,-1) 2 2( ) -2( )- 1 (2,3) 2 3 We can guess that is a parabola and observe that all over the x values, it has only one corresponding y value. So IT IS A FUNCTION. We can also perform the vertical line test and verify that it is a function because it crosses the curve a only one point.

y x (x-axis,y-axis) -1 1 2 2( ) -2( )- 1 2( ) -2( )- 1 2( ) -2( )- 1 Evaluate the following equation for the given domain and then determine if it represents a function. x = 2y - 2y - 1 2 D={-1, 0, 1, 2} 4 2 6 -2 -4 -6 8 10 -8 -10 x y y x (x-axis,y-axis) -1 1 2 2y - 2y - 1 2 2 2( ) -2( )- 1 3 -1 (3,-1) 2 2( ) -2( )- 1 -1 (-1,0) 2 2( ) -2( )- 1 -1 (-1,1) 1 2 2( ) -2( )- 1 3 (3,2) 2 We can guess that is a parabola that opens to the right but most values in the domain have two values in the range. So it is not a FUNCTION. We can also perform the vertical line test and verify that it is NOT a function because it crosses the curve at MORE THAN ONE POINT.

DISCRETE VS CONTINUOUS (-2,2), (2,5), (4,5), (8,-3), (10,8) f(x) = x +3 This function works for any real values in the domain. It is a CONTINUOUS FUNCTION. This is a function that only exist for the given values of its domain. IT IS A DISCRETE FUNCTION. 4 2 6 -2 -4 -6 8 10 -8 -10 x y 4 2 6 -2 -4 -6 8 10 -8 -10 x y

A function like f(x)=2x+1 Is a called a linear function. Linear functions are of the form y=mx+b. m and b are constants. The graph of a linear function is a line.

LINEAR Functions (Equations) Linear equations: Not linear equation: y = 2x + 5 1 1 4x – 3x + 2x – 3 = y 3 4 2 Exponents are different than 1. 5x- 2y = 6 1 1 5x = 2x + 5 1 1 2x - 4 = 4y + 6 1 1 1 1 4b + a = 10 y = 10 1 All are one or two variable equations with exponents always 1.

Is this relation a function?

Is this relation a function?

Is this relation a function?

Is this relation a function?

Homework 17-47 odd, 49, 52