1.8 Combinations of Functions

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1.8 Combinations of Functions

• JMerrill, 2010

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Arithmetic Combinations
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Sum

• Let
• Find (f g)(x)

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Difference

• Let
• Find (f - g)(x)

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Product

• Let
• Find

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Quotient

• Let
• Find

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You Do Let

• Find(fg)(x)
• (fg)(x)
• (f-g)(x)
• (g-f)(x)

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Finding the Domain of Quotients of Functions

• To find the domain of the quotient, first you
  must find the domain of each function. The
  domain of the quotient is the overlap of the
  domains.

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Example

• The domain of f(x)
• The domain of g(x) -2,2

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Example

• Since the domains are f(x)
• g(x)
  -2,2
• The domains of the quotients are

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Composition of Functions

• Most situations are not modeled by simple linear
  equations. Some are based on a system of
  functions, others are based on a composition of
  functions.
• A composition of functions is when the output of
  one function depends on the input from another
  function.

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Compositions Cont

• For example, the amount you pay on your income
  tax depends on the amount of adjusted gross
  income (on your Form 1040), which, in turn,
  depends on your annual earnings.

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Composition Example

• In chemistry, the process to convert Fahrenheit
  temperatures to Kelvin units
• This 2-step process that uses the output of the
  first function as the input of the second
  function.

This formula gives the Celsius temp. that
corresponds to the Fahrenheit temp.
This formula converts the Celsius temp. to Kelvins
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Composition Notation

• (f o g)(x) means f(g(x))
• (g o f)(x) means g(f(x)

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Composition of Functions A Graphing Approach
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You Do

• f(g(0))
• g(f(0))
• (fg)(3)
• (fg)(-3)
• (gf)(4)
• (fg)(4)

4
4
f(x)
3
3
g(x)
0
0
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Compositions Algebraically

• Given f(x) 3x2 and g(x) 5x1
• Find f(g(2)) Find g(f(4))
• g(2)5(2)1 11
• f(11) 3(11)2
• 363

How much is f(4)? g(48) 5(48)1241
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Compositions Algebraically Cont

• Given f(x) 3x2 and g(x) 5x1
• Find f(g(x)) Find g(f(x))
• What does g(x)?
• f(5x1)
• 3(5x1)2
• 3(25x210x1)
• 75x230x3

What does f(x)? g(3x2) 5(3x2)115x21
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You Do

• f(x)4x2-1 g(x) 3x
• Find