Inverse Functions

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Inverse Functions
NOTE f-1 (x) does NOT mean 1/f(x)
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Inverse Functions

• Example Verify that f(x)-3x6 and g(x)-1/3x2
  are inverses
• Solution Find f(g(x)) and g(f(x)). If they
  both equal x, then they are inverses.

g(f(x)) -1/3(-3x6)2 x-22 x
f(g(x)) -3(-1/3x2)6 x-66 x
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ONE-TO-ONE FUNCTIONS
A function is one-to-one if for each y-value
there is only one x-value that can be paired with
it that is, no two ordered pairs have the same
second component.
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Horizontal Line Test

• Used to determine whether a functions inverse
  will be a function by seeing if the original
  function passes the horizontal line test.
• If the original function passes the horizontal
  line test, then its inverse is a function.
• If the original function does not pass the
  horizontal line test, then its inverse is not a
  function.

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Finding the Inverse of a function

• The inverse of a function f can be found as
  follows
• Replace f (x) with y.
• Interchange x and y.
• Solve the resulting equation for y. If this
  equation does not define y as a function of x,
  the function does not have an inverse. Or
  equivalently, the function is f is not a
  one-to-one function and has no inverse
• If f has an inverse, replace y in step 3 with f
  -1 (x).

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Sine Function Cosine Function
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Inverse Trig Functions

• Goals
• Understand Domain/Range requirements for inverse
  trig functions
• Be able to calculate exact values for inverse
  trig functions
• Understand the composition of inverse trig
  functions

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The Inverse Sine Function

• The inverse sine function, also called the
  arcsine function, is denoted by sin-1 or arcsin.
• The function sin-1 has domain -1,1 and range
  -p/2, p / 2
• In other words, y arcsin x, iff sin y x

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The Inverse Cosine Function

• The inverse cosine function, also called the
  arccosine function, is denoted by cos-1 or
  arccos.
• The function cos-1 has domain -1,1 and range
  0, p
• In other words, if y arccos x, iff cos y x

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The Inverse Tangent Function

• The inverse tan function, also called the arctan
  function, is denoted by tan-1 or arctan.
• The function tan-1 has domain of all real numbers
  and range -p/2 y p/2
• In other words, y arctan x, iff tan y x

tan (tan -1 x ) x for real numbers
tan -1(tan x ) x -p/2 x p/2
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Summary

• With no restrictions, inverses of sin, cos and
  tangent are not functions.
• With restrictions
• Arcsin(x) ?
• Domain x-1 x 1
• Range? -p/2 ? p/2
• Arccos(x) ?
• Domain x-1 x 1
• Range ?0 ? p
• Arctan(x) ?
• Domain xx ? R
• Range ? -p/2 ? p/2