CCSS.Math.Content.HSF-BF.B.4b
The standard
(+) Verify by composition that one function is the inverse of another.
Common Core State Standards for Mathematics · Build new functions from existing functions
What this standard means
Students need to prove two functions undo each other by composing them both ways. They should show that f(g(x)) = x and g(f(x)) = x, while also paying attention to any domain restrictions that make the inverse relationship true.
Mastery looks like clean substitution, correct simplification, and a clear conclusion about whether the functions are inverses. Students often get stuck by checking only one composition, losing track of parentheses, or ignoring restricted domains, especially with square roots, quadratics, and rational functions.
Ways to teach it
- Hands-on activity: Give pairs matching cards with functions and possible inverses, then have them verify matches by composing both ways on paper.
- Discussion prompt: Ask, "Why is checking only f(g(x)) not enough to prove two functions are inverses?"
- Quick assessment: Post two functions and have students compute both compositions, then write one sentence saying whether they are inverses.
- Real-world connection: Use Celsius and Fahrenheit conversion formulas and have students compose them to show each conversion undoes the other.
Plan a lesson for CCSS.Math.Content.HSF-BF.B.4b
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Related standards
- CCSS.Math.Content.HSF-BF.B.4
Find inverse functions.
- CCSS.Math.Content.HSF-BF.A.1c
(+) Compose functions.
- CCSS.Math.Content.HSF-BF.B.4c
(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
- CCSS.Math.Content.HSF-BF.B.4d
(+) Produce an invertible function from a non-invertible function by restricting the domain.