Functions

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Functions
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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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M Mother Function
Joe Samantha Anna Ian Chelsea George
Laura Julie Hilary Barbara Sue
Humans
Mothers
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M Mother function

• Domain of M Joe, Samantha, Anna, Ian,
  Chelsea, George
• Range of M Laura, Julie, Hilary, Barbara
• In function notation we write
• M(Anna) Julie
• M(George) Barbara
• M(x)Hilary indicates that x Chelsea

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For the function f below , evaluate f at the
indicated values and find the domain and range of
f
f(1) f(2) f(3) f(4) f(5) f(6) f(7) Domain of
f Range of f

10 11 12 13 14 15 16
1 2 3 4 5 6 7
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Set Form of the Definition of a Function

• A function is a set of ordered pairs with the
  property that no two ordered pairs have the same
  first component and different second components.
• The set of all first components in a function is
  called the domain of the function, and the set of
  all second components is called the range.

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Determine which of the following relations
represent functions.
Not a function.
Function.
Function.
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The mother function M written as ordered pairs

• M (Joe, Laura), (Samantha, Laura),
• (Anna, Julie), (Ian, Julie), (Chelsea, Hillary),
• (George, Barbara)

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Graphical Displays of Functions

• Another way to depict a function whose ordered
  pairs are made up of numbers, is to display the
  ordered pairs via a graph on the coordinate
  plane, with the first elements of the ordered
  pairs graphed along the horizontal axis, and the
  second elements graphed along the vertical axis.

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The function f (-3,-1), (-2,-3), (-1,2),
(0,-1), (1,3), (2,4), (3,5) is graphed below.
Domain of f -3, -2, -1, 0, 1, 2, 3 Range of f
-3, -1, 2, 3, 4, 5
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Functions defined by Rules

• Let f be function, defined on the set of natural
  numbers, consisting of ordered pairs where the
  second element of the ordered pair is the square
  of the first element.
• Some of the ordered pairs in f are
• (1,1) (2,4), (3,9), (4,16),.
• f is best defined by the rule f(x) x²

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Function Notation f(x)

• Functions defined on infinite sets are denoted by
  algebraic rules.
• Examples of functions defined on all real numbers
  R.
• f(x) x² g(x) 2x - 1 h(x) x³
• The symbol f(x) represents the real number in
  the range of the function f corresponding to the
  domain value x.
• The ordered pair (x,f(x)) belongs to the
  function f.

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Evaluating functions
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Find the domain of the following functions
A)
B)
Domain is all real numbers but
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C)
Square root is real only for nonnegative numbers.
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Graph of a function

• The graph of the function f(x) is the set of
  points (x,y) in the xy-plane that satisfy the
  relation y f(x).
• Example The graph of the function
• f(x) 2x 1 is the graph of the equation
• y 2x 1, which is a line.

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Domain and Range from the Graph of a function
Domain x / or
Range y /
or
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Determine the domain, range, and intercepts of
the following graph.
y
4
(2, 3)
(10, 0)
0
(4, 0)
(1, 0)
x
(0, -3)
-4
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Theorem Vertical Line Test
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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y
x
Not a function.
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y
x
Function.
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Is this a graph of a function?
y
4
(2, 3)
(10, 0)
0
(4, 0)
(1, 0)
x
(0, -3)
-4
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Even functions

• A function f is an even function if
• for all values of x in the domain of f.
• Example is even
  because

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Odd functions
A function f is an odd function if for all
values of x in the domain of f. Example
is odd because
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Determine if the given functions are even or odd
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Graphs of Even and Odd functions

• The graph of an even function is symmetric with
  respect to the x-axis.
• The graph of an odd function is symmetric with
  respect to the origin.

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Determine if the function is even or odd?
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Determine if the function is even or odd?
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Determine if the function is even or odd?
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The sum fg is the function defined by
(f g)(x) f(x) g(x)
The domain of fg consists of numbers x that are
in the domain of both f and g.
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The difference f-g is the function defined by
(f - g)(x) f(x) - g(x)
The domain of f-g consists of numbers x that are
in the domain of both f and g.
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The product f g is the function defined by
(f g)(x) f(x) g(x)
The domain of f g consists of numbers x that are
in the domain of both f and g.
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Let
Find