Lesson 5 of 6
Find an inverse and state the domain that makes it a function.
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Big ideas · 1 of 9
Big ideas
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An Inverse functionA rule that reverses a one-to-one function.Exactly: The inverse of a one-to-one function reverses its input-output correspondence: on the domain of , and on its range.Mix-up alert: The inverse function is not the reciprocal .See this in Key words reverses input and output. If , then when the inverse exists on that range.
A function is one-to-one if different inputs give different outputs. Without this property, reversing pairs can give multiple outputs for one input.
To find an inverse rule, write , swap input and output roles, then solve for the new output. Verify by composition.
The original range becomes the inverse domain. Restricting an original domain can make an inverse possible; state the restriction with the formula.
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Worked example: reverse a linear rule
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Both domains and ranges are all real numbers.
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Without the restriction, squaring would lose the sign and would not be one-to-one.
Do this: Explain why the tempting shortcut fails.
Wrong idea: inverse notation means . An inverse undoes input-output pairing; a reciprocal divides 1 by an output. For , the inverse is x−2, not .
Worked example
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Worked example
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Thus .
Tip
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Class handout reference · Unit 2, one-to-one functions section 2.7, pages 37-39; inverse functions section 2.8, pages 41-43. Page 40 is blank. Use your class copy.
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Reverse the input-output pair.
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If and f has an inverse, what is ?
2.
What does one-to-one mean?
Different inputs produce different outputs.
Which line test checks one-to-one behavior?
The horizontal line test on the function graph.
What is the inverse of ?
.
What is the inverse of ?
.
Why is not invertible on all real inputs?
Inputs x and −x give the same square when .
What is the inverse of restricted to ?
, with inverse domain .
How do domains and ranges change under inversion?
They exchange roles.
True or false
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Self-test
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6 questions. One at a time.
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Homework · 3 points
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Homework
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https://gradebridge-beryl.vercel.app/learn/precalculus/functions/inverse-functions#homework
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