Section 1'9 Inverse Functions

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Section 1.9 Inverse Functions
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What you should learn

• How to find inverse functions informally and
  verify that two functions are inverse functions
  of each other
• How to use graphs of functions to determine
  whether functions have inverse functions
• How to use the horizontal line test to determine
  if functions are one-to-one
• How to find inverse functions algebraically

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Consider the function, f(x) that doubles x and
then subtracts 4.
Now swap x and y to create a new function g(x)
This function needs to add 4 first, then divide
by 2
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Inverse Notation

• f -1(x) is read as the f-inverse
• Notice that an inverse of a function does the
  opposite thing in the opposite order of the
  original function.

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Composition of Inverses
What is the name of this function?
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Definition of Inverse Function

• Let f and g be two functions such that
• f(g(x))x
• for every x in the domain of g
• and
• g(f(x))x
• for every x in the domain of f.
• Under these conditions the function g is the
  inverse function of the function f.

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Verifying Inverse Functions

• To verify that two functions f and g are inverse
  functions, form the composition.
• If the composition yields the identity function
  then the two functions are inverses of each other.

Therefore g(x) is not the inverse of f(x).
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Graphs of Inverse Functions
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Vertical Line Test?
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Vertical Line Test?
The inverse relation does not pass the vertical
line test.
Not a function.
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Horizontal Line Test

• A function f has an inverse function if and only
  if no horizontal line intersects the graph of f
  at more than one point.

The inverse relation will not be a function.
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One-to-One

• A function f is one-to-one if each value of the
  dependent variable corresponds to exactly one
  value of the independent variable.
• A function f has in inverse function if and only
  if f is one-to-one.

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One-to-One?
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One-to-One?
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One-to-One?
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One-to-One?
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One-to-One?
FAIL Vertical Line Test
PASS Horizontal Line Test
NOT One-to-One
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One-to-One?
Pass Vertical Line Test
PASS Horizontal Line Test
One-to-One
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Finding Inverse Functions Algebraically

• Use the horizontal line test to decide whether f
  has an inverse function.
• Replace f(x) with y.
• Swap x and y, and solve for y.
• Replace y with f -1(x).
• Verify that the range of one is the domain of the
  other.

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Find f -1(x)
Replace f(x) with y
Swap x with y
Solve for y
Replace y with f -1 (x)