Composition of Functions

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Compositionof Functions
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Learning Goal

• I will be able to write and evaluate composite
  functions.

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You can perform operations on functions in much
the same way that you perform operations on
numbers or expressions. You can add, subtract,
multiply, or divide functions by operating on
their rules.
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Another function operation uses the output from
one function as the input for a second function.
This operation is called the composition of
functions.
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The order of function operations is the same as
the order of operations for numbers and
expressions. To find f(g(3)), evaluate g(3) first
and then substitute the result into f.
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Example 1 Evaluating Composite Functions
Given f(x) 2x and g(x) 7 x, find each value.
f(g(4))
Step 1 Find g(4)
g(4) 7 4
g(x) 7 x
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Step 2 Find f(3)
f(3) 23
f(x) 2x
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So f(g(4)) 8.
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You Try Evaluating Composite Functions
Given f(x) 2x and g(x) 7 x, find each value.
g(f(4))
Step 1 Find f(4)
f(4) 24
f(x) 2x
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Step 2 Find g(16)
g(16) 7 16
g(x) 7 x.
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So g(f(4)) 9.
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You can use algebraic expressions as well as
numbers as inputs into functions. To find a rule
for f(g(x)), substitute the rule for g into f.
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Example 2 Writing Composite Functions
f(g(x))
Substitute the rule g into f.
Use the rule for f. Note that x ? 1.
Simplify.
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Example 3 Writing Composite Functions
f(g(x))
Substitute the rule g into f.
Distribute. Note that x 0.
Simplify.
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You Try Writing Composite Functions
g(f(x))
g(f(x)) g(x2 1)
Substitute the rule f into g.
Use the rule for g.
Simplify. Note that x ? .
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You Try Writing Composite Functions
g(f(x))
Substitute the rule f into g.
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End of notes.Homework

• p. 686 9, 11, 13, 25, 27, 29, 31, 39, 41, 43