7'1 Inverse Functions

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7.1 Inverse Functions

• Math 6B
• Calculus II

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One to One Function

• A function f is called a one-to-one function if
  it never takes on the same value twice that is
• Horizontal Line Test
• A function is one-to-one if and only if no
  horizontal line intersects its graph more than
  once.

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Inverse Functions

• Let f be a one-to-one function with domain A and
  range B. Then its inverse function f-1 has
  domain B and range A and is defined by
• for any y in B.

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Inverse Functions

• domain of f -1 range of f
• range of f -1 domain of f
• f 1 does not mean 1/f (x)

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Inverse Functions
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How to Find the Inverse of a One-to-One Function

• Write y f (x)
• Solve this equation for x in terms of y (if
  possible)
• To express f 1 as a function of x, interchange x
  and y. The resulting equation is y f 1(x)

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The Calculus of Inverse Functions

• If f is a continuous function defined on an
  interval, then its inverse function f 1 is also
  continuous.

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The Calculus of Inverse Functions

• Let the point (a , b) be on the curve f 1(x).
  If we want to find the derivative of f 1at xa
  it is equivalent to finding the reciprocal of the
  derivative of f at x b.