The Inverse Function of the Exponential Function

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Section 8.6

• The Inverse Function of the Exponential Function

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Definition

• The number e is an irrational number that shows
  up a lot in the real world. It is approximately
  given by

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One-to-One Functions

• Impose a puritan ethic, each boy goes out with
  only one girl and each girl goes our with only
  one boy.

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One-to-One Functions

• A function is one-to-one if different inputs have
  different outputs. That is, if for any a ? b we
  have f(a) ? f(b). If a function is one-to-one
  then its inverse will also be a function.

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Examples

• Each state has a capital and every capital has a
  state. So, if I ask you what the capital of
  Alaska was
• Each state has 2 senators. So, Boxer goes with
  California and so does Feinstein.

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FUNCTION
f
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One-To-One FUNCTION
f
f -1
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Vertical Line Test for Functions

• If a vertical line intersects a graph once and
  only once for each element of the domain, then
  the graph is a function.

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Horizontal Line Test for One-to-One Functions

• If a horizontal line intersects a graph once and
  only once for each element of the range, then the
  graph is a one-to-one function.

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• Determine if the function is one-to-one.

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Inverse Functions

• If we want to reverse the process that a function
  resulted in, then we use the inverse function.
  Hersheys doesnt waste much.

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Inverse Function

• Consider the UNDO Reeses peanut butter cup
  machine.

Input
Output Chocolate and Peanut Butter
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• If a function is one-to-one then it will have an
  inverse function denoted f -1(x) Find the inverse
  functions of the following.

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Inverse of the y 2x

• Graph the following on the same set of axes

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What is the Inverse Function?

• English to the rescueThe power we raise 2 to
  get x is y

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Definition
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Examples

• Evaluate the following

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Convert

• Convert from exponential to logarithmic equations

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Convert

• Convert from logarithmic to exponential equations

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• Solve for x

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Since they are inverses
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Bases Besides 10 and e

• The change of base theorem

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Properties of Logs
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Examples From Homework
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