Lesson 2 of 6
Use one output as the next input and check the combined domain.
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Big ideas · 1 of 9
Big ideas
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A CompositionUse one function’s output as the next function’s input.Exactly: The composite function satisfies . It is defined when is in the domain of and is in the domain of .Mix-up alert: Composition is not multiplication, and order usually matters.See this in Key words f(g(x)) applies g first and f second. The output of the inside function becomes the input of the outside function.
When finding a composite rule, replace every input of the outside function with the whole inside expression, using parentheses.
f(g(x)) and g(f(x)) usually differ. Name the actual sequence of operations before choosing a rule for a model.
x must belong to the inside function's domain, and its output must be allowed by the outside function. Simplification does not erase these requirements.
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Reversing the order gives a different rule.
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All other real x are allowed.
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Wrong idea: f(g(x)) means f(x) times g(x). Composition feeds the inside output into the outside rule. Multiplication would be written f(x)g(x), with no nesting.
Worked example
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Worked example
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Exclude .
Tip
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Class handout reference · Unit 2, section 2.3, page 21. Use your class copy.
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Which function acts first in f(g(x))?
g acts first, then f.
If and , what is f(g(x))?
.
For those same functions, what is g(f(x))?
.
Does composition usually commute?
No; changing the order usually changes the rule.
What two conditions restrict a composite domain?
The inside input must be allowed, and its output must be allowed by the outside function.
Is f(x)g(x) composition?
No. It is multiplication of two outputs.
How can you decompose ?
Use , then .
How do you check a composite value?
Calculate the two stages separately and compare with the composite rule.
True or false
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Self-test
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Self-test
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6 questions. One at a time.
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Write it
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Homework · 3 points
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https://gradebridge-beryl.vercel.app/learn/precalculus/functions/composition#homework
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