3.2 Inverse Functions and Logarithms

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3.2 Inverse Functions and Logarithms
3.3 Derivatives of Logarithmic and Exponential
functions
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One-to-one functions

• Definition A function f is called a one-to-one
  function if it never takes on the same value
  twice that is
• f(x1) ? f(x2) whenever x1 ? x2.
• Horizontal line test A function f is one-to-one
  if and only if no horizontal line intersects its
  graph more than once.
• Examples f(x) x3 is one-to-one
• but f(x) x2 is not.

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Inverse functions

• Definition Let f be a one-to-one function with
  domain A and range B. Then the inverse function f
  -1 has domain B and range A and is defined by
• for any y in B.
• Note f -1(x) does not mean 1 / f(x) .
• Example The inverse of f(x) x3 is f -1(x)x1/3
  
• Cancellation equations

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How to find the inverse function of a one-to-one
function f

• Step 1 Write yf(x)
• Step 2 Solve this equation for x in terms of y
  (if possible)
• Step 3 To express f -1 as a function of x,
  interchange x and y. The resulting equation is y
  f -1(x)
• Example Find the inverse of f(x) 5 - x3

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Another example
Solve for x
Inverse functions are reflections about y x.
Switch x and y
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Derivative of inverse function
First consider an example
At x 2
At x 4
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Calculus of inverse functions

• Theorem If f is a one-to-one continuous function
  defined on an interval then its inverse function
  f -1 is also continuous.
• Theorem If f is a one-to-one differentiable
  function with inverse function f -1 and f ' (f -1
  (a)) ? 0, then the inverse function is
  differentiable and
• Example Find (f -1 )' (1) for f(x) x3 x 1
  
• Solution By inspection f(0)1, thus f -1(1)
  0
• Then

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Logarithmic Functions
Consider where agt0 and a?1
This is a one-to-one function, therefore it has
an inverse.
The inverse is called the logarithmic function
with base a.
Example
The most commonly used bases for logs are 10
and e
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Properties of Logarithms
Since logs and exponentiation are inverse
functions, they un-do each other.
Product rule
Quotient rule
Power rule
Change of base formula
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Derivatives of Logarithmic and Exponential
functions
Examples on the board.
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Logarithmic Differentiation

• The calculation of derivatives of complicated
  functions involving products, quotients, or
  powers can often be simplified by taking
  logarithms.
• Step 1 Take natural logarithms of both sides of
  an equation y f (x) and use the properties of
  logarithms to simplify.
• Step 2 Differentiate implicitly with respect to
  x
• Step 3 Solve the resulting equation for y'
• Examples on the board