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Functions and Functions and their Graphstheir Graphs
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RelationsRelations• 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
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Input Output
-3 3
1 -2
4 1
4
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!!!
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Identify the Domain and Range. Then Identify the Domain and Range. Then
tell if the relation is a function.tell if the relation is a function.Input Output
-3 3
1 1
3 -2
4
Domain = {-3, 1,3,4}Range = {3,1,-2}
Function?Yes: each input is mappedonto exactly one output
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Vertical Line TestVertical 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
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(-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.
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(-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
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Graphing and Evaluating Graphing and Evaluating
FunctionsFunctions
• 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
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• 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.
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Graphing an equation in 2 Graphing an equation in 2
variablesvariables
1.Construct a table of values
2.Graph enough solutions to recognize a pattern
3.Connect the points with a line or curve
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Graph: f(x)=y = Graph: f(x)=y = x + 1x + 1
Step 1Table of values
Step2:Step 3:
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PracticePracticeCreate 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
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More Practice!More Practice!1. Given f(x) = 9x - 1, find f(0).
2. If h(x) = -3, find x in h(x) = 7x + 4
3. 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?
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1. f(0))=9(0) – 1 = - 1 2. If h(x) = -3, in h(x) = 7x + 4, then -3 = 7x + 4 Solve for x and x = -1 3. Mary’s salary: y = 3,000 + 200x (x the number
of machines she sells, y her monthly salary) If Mary sells x = 11 machines, then y = 3,000 + 200(11) = $5,2004. a) y = 350 + 150m (m = month, y = money in savings account)b) 2000 = 350 + 150m and solve for m; m = 11 months