Sullivan Algebra and Trigonometry: Section 3.1

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Sullivan Algebra and Trigonometry Section 3.1

• Objectives
• Determine Whether a Relation Represents a
  Function
• Find the Value of a Function
• Find the Domain of a Function
• Identify the Graph of a Function
• Obtain Information from or about the Graph of a
  Function

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Let X and Y be two nonempty sets of real numbers.
A function from X into Y is a rule or a
correspondence that associates with each element
of X a unique element of Y.
The set X is called the domain of the function.
For each element x in X, the corresponding
element y in Y is called the image of x. The set
of all images of the elements of the domain is
called the range of the function.
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f
x
y
x
y
x
X
Y
RANGE
DOMAIN
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Example Which of the following relations are
function?
(1, 1), (2, 4), (3, 9), (-3, 9)
A Function
(1, 1), (1, -1), (2, 4), (4, 9)
Not A Function
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Functions are often denoted by letters such as f,
F, g, G, and others. The symbol f(x), read f of
x or f at x, is the number that results when x
is given and the function f is applied.
Elements of the domain, x, can be though of as
input and the result obtained when the function
is applied can be though of as output.
Restrictions on this input/output machine 1.
It only accepts numbers from the domain of the
function. 2. For each input, there is exactly
one output (which may be repeated for
different inputs).
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Example Given the function
Find
f (x) is the number that results when the number
x is applied to the rule for f.
Find
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The domain of a function f is the set of real
numbers such that the rule of the function makes
sense.
Domain can also be thought of as the set of all
possible input for the function machine.
Example Find the domain of the following
function
Domain All real numbers
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Example Find the domain of the following
function
Example Find the domain of the following
function
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When a function is defined by an equation in x
and y, the graph of the function is the graph of
the equation, that is, the set of all points
(x,y) in the xy-plane that satisfies the equation.
Vertical Line Test for Functions A set of
points in the xy-plane is the graph of a function
if and only if a vertical line intersects the
graph in at most one point.
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Example Does the following graph represent a
function?
y
x
The graph does not represent a function, since it
does not pass the vertical line test.
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Example Does the following graph represent a
function?
y
x
The graph does represent a function, since it
does passes the vertical line test.
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Determine the domain, range, and intercepts of
the following graph.
y
(2, 3)
4
(10, 0)
(4, 0)
0
x
(1, 0)
(0, -3)
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