1.6 Relations and Functions. Warm Up Use the graph for Problems 1–2. 1. List the x-coordinates of...
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Transcript of 1.6 Relations and Functions. Warm Up Use the graph for Problems 1–2. 1. List the x-coordinates of...
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1.6 Relations and Functions
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Identify the domain and range of relations and functions.
Determine whether a relation is a function.
Objectives
Vocabulary
relationdomainrangefunction
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relation • pairing of input values with output values• set of ordered pairs (x,y), where x is an input and y is an output
domain•set of input values for a relation
range•set of output values
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A
B
C
2
Domain Range
Mapping Diagram
Set of Ordered Pairs
{(2, A), (2, B), (2, C)}
(x, y) (input, output) (domain, range)
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Give the domain and range for each relation:
{(100,5), (120,5), (140,6), (160,6), (180,12)}
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Not a function: The relationship from number to letter is not a function because the domain value 2 is mapped to the range values A, B, and C.
Function: The relationship from letter to number is a function because each letter in the domain is mapped to only one number in the range.
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Determine whether each relation is a function.
Give the domain and range.A. from the items in a store to their prices on a
certain date
B. from types of fruits to their colors
C.
D. from the number of items in a grocery cart to the total cost of the items in the cart
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Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through.
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1.7 Function Notation
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Write functions using function notation.
Evaluate and graph functions.
Objectives
Vocabulary
function notationdependent variableindependent variable
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ƒ(x) = 5x + 3 ƒ(1) = 5(1) + 3
Output value Output value Input valueInput value
ƒ of x equals 5 times x plus 3. ƒ of 1 equals 5 times 1 plus 3.
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•f(x) is not “f times x” or
“f multiplied by x.”
•f(x) is pronounced “f of x”
•f(x) means “the value of the function at x.”
So f(1) represents the value of y at x =1
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To find the range given the domain
Substitute each value of the domain intothe function and solve for f(x).
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Example
ƒ(x) = 8 + 4x
For each function, evaluate ƒ(0), ƒ , and ƒ(–2).
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Example
For each function, evaluate ƒ(0), ƒ , and ƒ(–2).
ƒ(x) = x2 – 4x
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For each function, evaluate ƒ(0), ƒ , and ƒ(–2).
Use the graph to find the corresponding y-value for each x-value.
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To find the domain given the range
Substitute each value of the range into f(x).You then have an equation.Solve for x.
ƒ(x) = 8 + 4x
EX: Find the domain given the range, y=16
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In the notation ƒ(x), ƒ is the name of the function. The output ƒ(x) of a function is called the dependent variable because it depends on the input value of the function.
The input x is called the independent variable.
When a function is graphed, the independent variable is graphed on the horizontal axis and the dependent variable is graphed on the vertical axis.
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DISCRETE Functions
{(0, 4), (1, 5), (2, 6), (3, 7), (4, 8)}
Graph the points.
Do not connect the points because the values between the given points have not been defined.
A function whose graph is made up of unconnected points is called a discrete function.
What is the domain and range ofthis discrete function?
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A function whose graph is made up of connected points is called a continuous function.
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Domains and ranges of continuousfunctions are inequality statementsor R.
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The algebraic expression used to define a function is called the function rule.
The function described by f(x) = 5x + 3 is defined by the function rule 5x + 3.
To write a function rule, first identify the independent and dependent variables.
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Example
A carnival charges a $5 entrance fee and $2 per ride.
Write a function to represent the total cost after taking a certain number of rides.
Let r be the number of rides and let C be the total cost in dollars. The entrance fee is constant.
C(r) = 5 + 2r
First, identify the independent and dependent variables.
Cost depends on the entrance fee plus the number of rides taken
Cost = entrance fee + number of rides taken
Replace the words with expressions.
Dependent variable Independent variable
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What is the value of the function for an input of 12, and what does it represent?
Substitute 12 for r and simplify.
The value of the function for an input of 12 is 29. This means that it costs $29 to enter the carnival and take 12 rides.
C(12) = 5 + 2(12)
C(12) = 29
Example (cont.)
A carnival charges a $5 entrance fee and $2 per ride.