Chapter 8 Basics of Functions

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Chapter 8Basics of Functions
Definition Let A and be B two nonempty sets.
Then an assignment f from A to B is called a
function from A to B iff every element of A is
assigned a unique element of B.
Functions can be represented in many ways. We
can 1) describe them in words, 2) draw
Venn-diagrams 3) represent them using
equations, or 4) graph them.
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Chapter 8Basics of Functions
Definition Given a function f from A to B, we
define the set A to be the domain of f. The
domain is the collection of all input values of
f. For an element a in A, we say that f(a) is the
value of f at a. We say that a is the input and
f(a) is the output of the function f. The
collection of all output values is the set that
we call the range of f. The range is a subset of
B.
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EXAMPLE Basic Function
Determine whether the following is a function.
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drums
guitar
2
violin
3
flute
4
Domain
Range
SOLUTION
Yes, because none of the members of the domain
correspond to more than one member of the range.
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EXAMPLE Basic Functions
Determine whether the following is a function.
3
ants
beetles
8
crickets
5
moths
9
Domain
Range
SOLUTION
No, because one of the members of the domain, 9,
corresponds to more than one member of the range.
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EXAMPLE Basic Functions
Find f(3)
SOLUTION
Replace x with 3
Evaluate the exponent
Multiply
Add and Subtract
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EXAMPLE Basic Functions
Find Q(-4)
SOLUTION
Replace z with -4
Evaluate the exponents
Multiply
Add
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EXAMPLE Basic Functions
Find L(xy).
SOLUTION
Replace w with xy
Rewrite exponent
Multiply
Add
Distribute
NOTE THIS CANNOT BE SIMPLIFIED ANY FURTHER!!!
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The Vertical Line Test
Example Use the vertical line test to identify
graphs in which y is a function of x.
(a)
(b)
(c)
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The Vertical Line Test
SOLUTION
(a)
(b)
(c)
y is a function of x
y is not a function of x
y is a function of x
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Graphs of Functions
Example The figure shows the cost of mailing a
first-class letter, f(x), as a function of its
weight, x, in ounces. Use the graph to answer
the following questions.
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Graphs of Functions continued.
(a) Find f (3). What does this mean in terms of
the variables in this situation?
(b) What is the cost of mailing a letter that
weighs 1.5 ounces?
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Graphs of Functions continued.
SOLUTION
(a) Find f (3). What does this mean in terms of
the variables in this situation?
f (3) 0.83. This means that when a first-class
letter weighs 3 ounces, postage costs 83 cents.
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Graphs of Functions continued.
(b) What is the cost of mailing a letter that
weighs 1.5 ounces?
f (1.5) 0.60. This means that when a
first-class letter weighs 1.5 ounces, postage
costs 0.60.
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Graphs of Functions
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Domain and Range
EXAMPLE
Use the graph of the function to identify its
domain and range.
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7
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Domain and Range
SOLUTION
5
7
x x is a real number
Domain
y y 3.6
Range
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8.1 Introduction to Functions
The notes for this sections were created using
Blitzers Introductory and Intermediate Algebra.