Lesson 1.3, page 154 More on Functions

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Lesson 1.3, page 154More on Functions

• Objectives
• To find the difference quotient.
• Understand and use piecewise functions
• Identify intervals on which a function increases,
  decreases, or is constant.
• Use graphs to locate relative maxima or minima.
• Identify even or odd functions recognize the
  symmetries.
• Graph step functions.

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REVIEW of Lesson 1.2Reminder Domain
Restrictions

• For FRACTIONS
• No zero in denominator!
• For EVEN ROOTS
• No negative under even root!

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Find the domain of each (algebraically)and write
in interval notation.
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Functions Difference Quotients

• Useful in discussing the rate of change of
  function over a period of time
• EXTREMELY important in calculus
• (h represents the difference in two x values)
• DIFFERENCE QUOTIENT FORMULA

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Difference QuotientThe average rate of change
(the slope of the secant line)
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If f(x) -2x2 x 5, find and simplify each
expression.

• A) f(xh)

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If f(x) -2x2 x 5, find and simplify each
expression.

• B)

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Your turn Find the difference quotient f(x)
2x2 2x 1
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PIECEWISE FUNCTIONS

• Piecewise function A function that is defined
  differently for different parts of the domain a
  function composed of different pieces
• Note Each piece is like a separate function
  with its own domain values.
• Examples You are paid 10/hr for work up to 40
  hrs/wk and then time and a half for overtime.

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See Example 3, page 169.

• Check Point 2
• Use the function
• to find and interpret each of the folllowing
• a) C(40) b) C(80)

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Graphing Piecewise Functions

• Draw the first graph on the coordinate plane.
• Be sure to note where the inequality starts and
  stops. (the interval)
• Erase any part of the graph that isnt within
  that interval.

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Graph

See p.1015 for more problems.
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Describing the Function

• A function is described by intervals, using its
  domain, in terms of x-values.
• Remember

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Increasing and Decreasing Functions

• Increasing Graph goes up as you move from left
  to right.
• Decreasing Graph goes down as you move from
  left to right.
• Constant Graph remains horizontal as you move
  from left to right.

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Increasing and Decreasing
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Constant
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Increasing and Decreasing
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See Example 1, page 166.

• Check Point 1 See middle of
• page 166.

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Find the Intervals on the Domain in which the
Function is Increasing, Decreasing, and/or
Constant
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Relative Maxima and Minima

• based on y values
• maximum peak or highest value
• minimum valley or lowest value

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Relative Maxima and Relative Minima
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Even Odd Functions Symmetry

• Even functions are those that are mirrored
  through the y-axis. (If x replaces x, the y
  value remains the same.) (i.e. 1st quadrant
  reflects into the 2nd quadrant)
• Odd functions are those that are mirrored through
  the origin. (If x replaces x, the y value
  becomes y.) (i.e. 1st quadrant reflects into the
  3rd quadrant or over the origin)

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See Example 2, page 167.

• Determine whether each function is even, odd, or
  neither.
• a) f(x) x2 6 b) g(x) 7x3 - x

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Determine whether each function is even, odd, or
neither.

• c) h(x) x5 1

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Your turn Determine if the function is even,
odd, or neither.

1. Even
2. Odd
3. Neither