Back to Verify inverse by composition — Problem 4 · Task Set 41

HSF.BF.B.4.b Tutor Intake — Verifying Inverses by Composition

This short check helps your tutor understand where to start. Answer each question on your own. If you are not sure, give your best try — every response helps your tutor plan your sessions.

Grade 9·11 problems·~14 min·Common Core Math - HS Functions·standard·hsf-bf-b-4b

Work through problems with immediate feedback

Procedures

Let $f(x) = 2x + 1$ and $g(x) = \dfrac{x - 1}{2}$ .

A student performs a numeric check with $a = 3$ :

$f(g(3)) = f(1) = 3$ ✓
$g(f(3)) = g(7) = 3$ ✓

What can the student conclude from this numeric check?

$f$ and $g$ are inverses — both compositions equal 3

The numeric check is encouraging but does not prove the inverse relationship;
symbolic verification is still needed

Checking two compositions at one value is enough to prove inverses for all $x$

If a numeric check passes, symbolic verification will always confirm inverses,
so symbolic work is unnecessary