Functions

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Functions
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Representations of Functions

• There are four possible ways to represent a
  function
• verbally (by a description in words)
• numerically (by a table of values)
• visually (by a graph)
• algebraically (by an explicit formula)
• A The area of a circle depends on the
  radius of the circle. The rule that connects
  and is given by the equation
• With each positive number there is
  associated one value of and we say that
  is a function of

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B
C
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The rule that the U. S. Postal Service used as of
2001 is as follows The cost is 34 cents for up
to one ounce plus 22 cents for each successive
ounce up to 11 ounces.
D
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Graphs of Functions

• The graph of a function is a curve in the
  -plane. But the question arises Which curves in
  the -plane are graphs of functions This is
  answered by the following test.
• The Vertical Line Test A curve in the
  -plane is the graph of a function of if and
  only if no vertical line intersects the curve
  more than once.

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Classification of Functions

• We may classify functions by their formula as
  follows
• Polynomials
• Linear Functions Quadratic Functions.
  Cubic Functions.
• Piecewise Defined Functions
• Absolute Value Functions Step Functions
• Rational Functions
• Algebraic Functions
• Trigonometric and Inverse trigonometric
  Functions
• Exponential Functions
• Logarithmic Functions

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Functions Properties

• We may classify functions by some of their
  properties as follows
• Injective (One to One) Functions
• Surjective (Onto) Functions
• Odd or Even Functions
• Periodic Functions
• Increasing and Decreasing Functions
• Continuous Functions
• Differentiable Functions

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Symmetry
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Transformations of Functions
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Combinations of Functions
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Composition of Functions
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Power Functions
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Exponential Functions
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Inverse Functions
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Logarithmic Functions
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The logarithm with base is called the natural
logarithm and has a special notation
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When we try to find the inverse trigonometric
functions we have a slight difficulty. Because
the trigonometric functions are not one-to-one
they dont have inverse functions. The difficulty
is overcome by restricting the domains of these
functions so that hey become one-to-one.
Inverse Trigonometric Functions
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