Composite functions

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Composite functions

• Finding the composition of two functions.

Inverse functions

• Finding the inverse of a function.
• Exploring why some inverse functions dont exist.

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What is it?

• COMPOSITE FUNCTION - two functions put together.
• gf means f then g..

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WHAT??

• f(x)x2
• g(x)5x
• Calculate fg(2)

2
5 x 2
10
10
102
12
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Finding composite functions
Find fg(x)
Finally, this simplifies to
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• Im thinking of a number, I double it then add
  two. Write an expression for this.
• Im thinking of a number, I add two and then
  square it. Write an expression for this.
• What if the second number I think of is the
  output of the first. Can you write an expression
  for this?

Label the first expression f(x) and the second
expression g(x). Find gf(x) fg(x)
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Inverse functions

• Finding the inverse of a function.
• Exploring why some inverse functions dont exist.

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What is it?

• INVERSE FUNCTION reversing a function,
  undoing.
• f -1 notates an inverse function. (not 1/f)

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WHAT??

• Find the inverse of f(x)4x-2

4
-2
x
4x-2
(x2)/4
x
/4
2
So f -1 (x)(x2)/4
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A neat little trick

• As always in maths, there is a trick to this
• Write function as a rule in terms of y and x.
• Swap x and y
• Rearrange to get in terms of y.

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A neat little trick
Find the inverse of
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Reminder..
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Inverse functions

• What type of inverse function would a one-one
  function give? Is it still a function?
• What type of inverse function would a many-one
  function give? Is it still a function?

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Inverse functions

• Inverse functions only exist for one-one
  functions.

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Many-one functions
f(x)x2 has two inputs (-2 and 2) that generate
the output 4.
The inverse would give two possible outputs for
input of 4.
Functions cant have two possible outputs so
inverse DOES NOT exist.
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Things to note..

• The domain of f-1 is the range of f.
• The graph of an inverse function can be found by
  reflecting a function in the line yx
• Check this by plotting y3x1 and
• y(x-1)/3 on your graphic calculator.

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Reflecting..