Functions and their Graphs

Relations A relation is a mapping of input values with output values. The set of x-values (input values) is called the domain. The set of y-values (output values) is called the range. A relation is a function provided there is exactly one output for each input. Each element of the domain is paired with only one element of the range It is NOT a function if at least one input has more than one output

Input Output 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 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

Vertical Line Test You can use the vertical line test to visually determine if a relation is a function. Slide any vertical line 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

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

Graphing and Evaluating Functions Many functions can be represented by an equation in 2 variables: y = 2x - 7 or! f(x) = 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 = f(x) = 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 f(x) = 2x-7 ….. x is the independent variable and y is the dependent variable 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: f(x)=y = x + 1 Step 1 Table of values Step2: Step 3:

Practice Create a table with 5 different values. Graph the lines on the coordinate plane. 1.f(x) = 2x+ 3 2.h(x) = - 3x+1 3.g(x)= 5 – x 2.R(x)= x - 4

More Practice! 1.Given f(x) = 9x - 1, find f(0). 2.If h(x) = -3, find x in h(x) = 7x Mary is machine saleswoman who earns a base salary of $3,000 plus a commission $200 for every machine she sells. Write a functions (equation) that shows the total income Mary earns if she sells x machines in one month. How much money will Mary make in April if she sells 11 machines? 4.Paul opens a savings account with $350 dollars. He saves $150 per month. Assume that he does not withdraw money or make any additional deposits. a). Write a linear model that represents the total amount of money Paul has in his account after m months. b). After how many months will Paul have more than $2,000?

1.f(0))=9(0) – 1 = If h(x) = -3, in h(x) = 7x + 4, then -3 = 7x + 4 Solve for x and x = -1 3.Mary’s salary: y = 3, x (x the number of machines she sells, y her monthly salary) If Mary sells x = 11 machines, then y = 3, (11) = $5, a) y = m (m = month, y = money in savings account) b)2000 = m and solve for m; m = 11 months