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
Students
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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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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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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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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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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Increasing and Decreasing Functions

• A function f is increasing on an open interval I
  if, for any choice of x1 and x2 in I, with x1 lt
  x2 we have f(x1)ltf(x2).
• A function f is decreasing on an open interval I
  if, for any choice of x1 and x2 in I, with x1 lt
  x2 we have f(x1)gtf(x2).
• A function f is constant on an open interval I
  if, for all choices of x, the values f(x) are
  equal.

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Local Maximum, Local Minimum

• A function f has a local maximum at c if there is
  an open interval I containing c so that, for all
  x c in I, f(x) lt f(c). We call f(c) a local
  maximum of f.
• A function f has a local minimum at c if there is
  an open interval I containing c so that, for all
  x c in I, f(x) gt f(c). We call f(c) a local
  minimum of f.

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Determine where the following graph is
increasing, decreasing and constant. Find local
maxima and minima.
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y
4
(2, 3)
(4, 0)
0
(1, 0)
x
(10, -3)
(0, -3)
(7, -3)
-4