2.2 More on Functions and Their Graphs

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2.2 More on Functions and Their Graphs
See p 202 re divorce and years of marriage,
discuss trends
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Definition of a Difference Quotient

• The expression
• for h not equal to 0 is called the difference
  quotient.
• This will ABSOLUTELY BE ON YOUR TEST!
• FYI This is used to find the derivative (slope
  of a function for a very small (h) change in x)
  in calculus. Lets discuss how this is related to
  the graph for the divorce rate.

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Ex Find the difference quotient for the
following functions
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As you have seen, one type of function may
describe a trend for a certain period, but then
another type of function will describe that
function for a later period. For example, what
happens to your cell phone bill if you go over
your minutes? (Intro module on muscle tension.)
See p 214 formulas 73 if interested in a real
example.
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Evaluating Piecewise Defined Functions

• Determine to which piece your x-value
  (independent variable) belongs.
• Plug that x-value into that piece of the function
  and evaluate as usual.
• Laymans terms, see which ride the x may take
  or which back door you are permitted to enter.

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Ex Evaluate the piecewise function as
given x 5 if x gt -5 f(x) -(x 5) if
x lt -5 Find a) f(0) b) f(-6) c) f(-5) Now
graph the piecewise function.
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Increasing, Decreasing, and Constant Functions
A function is increasing on an open interval if
for any x1, and x2 in the interval, where x1 lt
x2, then f (x1) lt f (x2). A function is
decreasing on an open interval if for any x1, and
x2 in the interval, where x1 lt x2, then f (x1) gt
f (x2). A function is constant on an open
interval if for any x1, and x2 in the interval,
where x1 lt x2, then f (x1) f (x2).
Note read from left to right endpoints are
not included.
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EX Describe the increasing, decreasing, or
constant behavior of each function whose graph is
shown.
Solution

• The function is decreasing on the interval (
  , ), increasing on the interval ( ,
  ), and decreasing on the interval (
  , ).
• b. Although the function's equations are not
  given, the graph indicates that the function is
  defined in two pieces. The part of the graph to
  the left of the y-axis shows that the function is
  ____________on the interval (-oo, 0). The part to
  the right of the y-axis shows that the function
  is ________creasing on the interval (0, oo)
  (Note that the endpoint is not included).
• How would the intervals change if the arrow on
  the right were missing?

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Definitions of Relative Maximum and Relative
Minimum

• A function value f(a) is a relative maximum of f
  if there exists an open interval about a such
  that f(a) gt f(x) for all x in the open interval.
• A function value f(b) is a relative minimum of f
  if there exists an open interval about b such
  that f(b) lt f(x) for all x in the open interval.

Do p 213 62 together.