Chapter 2 Functions and Linear Functions

1
Chapter 2Functions and Linear Functions
2
2.1
Introduction to Functions
3
Relation
Definition of a Relation A relation is any set of
ordered pairs. The set of all first components
of the ordered pairs is called the domain of the
relation and the set of all second components
is called the range of the relation.
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Functions
Definition of a Function A function is a
correspondence from a first set, called the
domain, to a second set, called the range, such
that each element in the domain corresponds to
exactly one element in the range. (Each x
corresponds to exactly one y in a function.
Values of x have to be faithful so to speak
that is, each x goes to exactly one y. However,
a y value may be the image of more than one x
so ys are not required to be faithful) For
example, the following relation is a
function (1,2), (2,3), (3,5), (4,3) Note
that the y value 3 is the image of two x values.
Blitzer, Algebra for College Students, 6e Slide
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Basic Functions
EXAMPLE
Determine whether the following is a function.
1
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.
Blitzer, Algebra for College Students, 6e Slide
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Basic Functions
EXAMPLE
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.
Blitzer, Algebra for College Students, 6e Slide
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Functions
Example Determine whether each relation is a
function (a) (2,3), (4,5), (6,5), (9,10) (a)
Yes a function (b) (1,2), (3,3), (6,8),
(1,10) (b) No not a function since 1 is
mapped to 2 and 10 (c) (1,5), (4,5), (2,5),
(3,5) (c) Yes a function
Blitzer, Algebra for College Students, 6e Slide
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Functions as Equations
Functions are usually given in terms of equations
rather than as sets of ordered pairs. For
example, consider the function y 2x 3 For
each value of x, there is one and only one value
of y. The variable x is called the independent
variable and the variable y is called the
dependent variable.
Blitzer, Algebra for College Students, 6e Slide
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Function Notation
If an equation in x and y gives only one value of
y for each value of x, then the variable y is a
function of the variable x. When an equation
represents a function, the function is often
named by a letter such as f, g, h, F, G, or
H. The output value, the y value, is often
denoted by f(x), read f of x or f at x The
notation f(x) does not mean f times x. The
notation describes the value of the function f
at x.
Blitzer, Algebra for College Students, 6e Slide
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Basic Functions
EXAMPLE
Find the indicated function value
SOLUTION
Replace x with 3
Evaluate the exponent
Multiply
Add and Subtract
Blitzer, Algebra for College Students, 6e Slide
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Basic Functions
EXAMPLE
Find the indicated function value
SOLUTION
Replace z with -4
Evaluate the exponents
Multiply
Add
Blitzer, Algebra for College Students, 6e Slide
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Basic Functions
EXAMPLE
Find the indicated function value
SOLUTION
Replace w with xy
Rewrite exponent
Multiply
Add
Distribute
NOTE THIS CANNOT BE SIMPLIFIED ANY FURTHER!!!
Blitzer, Algebra for College Students, 6e Slide
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2.1 Assignment
p. 102 (2-22 even)