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
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students compose two functions in both orders and simplify each result. They check domain restrictions and decide whether each composition returns the original input.
What Mastery Looks Like
- A student accurately calculates f(g(x)) and g(f(x)) and simplifies both to x. The student also states any input restrictions that affect the conclusion.
Common Misconceptions
- Students may check only one composition instead of both orders. They may confuse an inverse function with a reciprocal, or ignore domain restrictions when simplifying.
How to Assess It
- Give: f(x) = 3x - 5 and g(x) = (x + 5)/3. Find both compositions and state whether the functions are inverses.
Lesson moves
Ways to Teach It
Give pairs function-machine cards, then have students pass an input through both machines and check whether it returns unchanged.
Ask students to explain why checking only f(g(x)) may not fully verify an inverse relationship.
Run an inverse matching game where students pair function cards, calculate both compositions, and reject pairs that do not return x.
Compose the Celsius-to-Fahrenheit and Fahrenheit-to-Celsius formulas to show that converting there and back returns the original temperature.
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Printable HSF-BF.B.4b Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSF-BF.B.4b, with an answer key for the teacher on its own page. No account needed.
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Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSF-IF.A.1
Verifying inverses by composition requires understanding functions as input-output rules and using notation like f(x) for outputs.
- CCSS.Math.Content.HSF-IF.A.2
Verifying inverses by composition requires reading function notation and evaluating one function at another function’s output.
- CCSS.Math.Content.HSF-BF.A.1c
Verifying inverse functions requires composing them in both orders and recognizing the identity output.
Keep exploring
Related Standards
- CCSS.Math.Content.HSF-BF.B.4
Find inverse 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.
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