12'1 Inverse Functions

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12.1 Inverse Functions

• For an inverse function to exist, the function
  must be one-to-one.
• One-to-one function each x-value corresponds to
  only one y-value and each y-value corresponds to
  only one x-value.
• Horizontal Line Test A function is one-to-one
  if every horizontal line intersects the graph of
  the function at most once.

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12.1 Inverse Functions

• f-1(x) the set of all ordered pairs of the form
  (y, x) where (x, y) belongs to the function f.
  Note
• Since x maps to y and then y maps back to x it
  follows that

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12.1 Inverse Functions

• Method for finding the equation of the inverse of
  a one-to-one function
• Interchange x and y.
• Solve for y.
• Replace y with f-1(x)

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12.1 Inverse Functions

• Example
• Interchange x and y.
• Solve for y.
• Replace y with f-1(x)

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12.1 Inverse Functions

• Graphing inverse functions The graph of an
  inverse function can be obtained by reflecting
  (getting the mirror image) of the original
  functions graph over the line y x

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12.2 Exponential Functions

• Exponential Function For a gt 0 and a not equal
  to 1, and all real numbers x,
• Graph of f(x) ax
• Graph goes through (0, 1)
• If a gt 1, graph rises from left to right. If 0 lt
  a lt 1, graph falls from left to right.
• Graph approaches the x-axis.
• Domain is Range is

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12.2 Exponential Functions Graph of an
Exponential Function
(0, 1)
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12.2 Exponential Functions

• Property for solving exponential equations
• Solving exponential equations
• Express each side of the equation as a power of
  the same base
• Simplify the exponents
• Set the exponents equal
• Solve the resulting equation

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12.2 Exponential Functions

• Example Solve 9x 27

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12.3 Logarithmic Functions

• Definition of logarithm
• Note logax and ax are inverse functions
• Since b1 b and b0 1, it follows thatlogb(b)
  1 and logb(1) 0

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12.3 Logarithmic Functions

• Logarithmic Function For a gt 0 and a not equal
  to 1, and all real numbers x,
• Graph of f(x) logax
• Graph goes through (0, 1)
• If a gt 1, graph rises from left to right. If 0 lt
  a lt 1, graph falls from left to right.
• Graph approaches the y-axis.
• Domain is Range is

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12.3 Logarithmic Functions Graph of an
Exponential Function
Try to imagine the inverse function
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12.3 Logarithmic Functions Inverse - Logarithmic
Function
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12.3 Logarithmic Functions

• Example Solve x log1255In exponential
  formIn powers of 5Setting the powers equal

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12.4 Properties of Logarithms

• If x, y, and b are positive real numbers
  whereProduct RuleQuotient RulePower Rule
  Special Properties

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12.4 Properties of Logarithms

• ExamplesProduct RuleQuotient RulePower
  Rule Special Properties

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13.1 Additional Graphs of Functions Absolute
Value Function

• Graph of
• What is the domain and the range?

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13.1 Additional Graphs of Functions Graph of a
Square Root Function

• Graph of

(0, 0)
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13.1 Additional Graphs of Functions Graph of a
Greatest Integer Function

• Graph ofGreatest integerthat is less than
  orequal to x

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13.1 Additional Graphs of Functions Shifting of
Graphs

• Vertical ShiftsThe graph is shifted upward by k
  units
• Horizontal shiftsThe graph is shifted h units
  to the right
• If a lt 0, the graph is inverted (flipped)
• If a gt 1, the graph is stretched (narrower) If 0
  lt a lt 1, the graph is flattened (wider)

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13.1 Additional Graphs of Functions

• Example GraphGreatest integerfunction
  shiftedup by 4

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13.1 Additional Graphs of Functions Composite
Functions

• Composite function function of a
  functionf(g(x)) (f ? g)(x)Example if f(x)
  2x 1 and g(x) x2 thenf(g(x)) f(x2) 2x2
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• What is g(f(2))?
• Does f(g(x)) g(f(x))?