Verify Inverses by Composition

(+) HSF.BF.B.4.b

In this lesson:

• Prove an inverse relationship with a two-direction composition test
• Verify symbolically, screen numerically, and use composition to catch errors

What You'll Be Able to Do

By the end of this lesson, you should be able to:

1. State the two-direction definition of inverses
2. Verify inverses by composing both directions
3. Explain why one direction isn't enough
4. Catch errors in a proposed inverse
5. Screen pairs numerically
6. Test pairs that only look inverse

How Do You Know Your Inverse Is Right?

You just learned to find an inverse algebraically. But finding an answer and
knowing it's correct are different things.

How do you actually prove two functions undo each other?

Quick Recall: What Does Mean?

means substitute everywhere has an input.

The Two-Direction Definition of Inverses

Two functions and are inverses if and only if:

Both directions are required. A pair is not certified until each direction
comes back clean.

A Pair That Passes Both Directions

Claim: A classmate says and are inverses. Check it.

Both Arrows Return to x

A certified pair sends every input back to itself, in both directions.

Predict First: Does This Pair Pass?

(all real numbers) and .

Does this pair pass both directions of the composition test?

• Predict: yes, both pass — or no, at least one fails

Commit to an answer before the next slide reveals it.

One Direction Passes. One Doesn't.

for , but for .

One Passing Direction Proves Nothing

and are not full inverses over all real numbers —
only on the restricted domain .

• A single passing direction can hide a failing one
• Certifying a pair requires checking both, every time

Quick Check: Is One Direction Enough?

If for all , are and definitely inverses?

Think it through before the next slide.

From Proof Standard to Procedure

The two-direction test is the standard. Now let's run it symbolically,
every time, on pairs of increasing complexity.

Symbolic Verification: Substitute and Simplify

• Substitute into every in ; simplify — should reach
• Substitute into every in ; simplify — should reach
• Each step is routine algebra you already know how to do

Worked Example: A Linear Pair

Your Turn: A Power Pair

Simplify the cube of a cube root — what's left?

Same Pattern, One More Time

(all reals),

Quick Check: Name the Shared Pattern

Why does the pair fail the same way the pair did? What's the pattern?

Name the shared cause before moving on.

Worked Example: A Rational Pair

The Rational Pair: The Other Direction

Both directions reach — the rational pair is certified.

Quick Check: The Trickiest Step

Which step in the rational example was trickiest, and why?

Name the specific step, not just "the whole thing."

Watch Out for Simplification Slips

A slip during simplification can fake a pass or hide a real pass as a fail.

• If unsure your simplified expression truly equals
• Substitute one number into the simplified expression and into
• Compare the two results directly

Your Turn: Two Verification Problems

1. Verify and are inverses
2. Verify and are inverses

Check both directions for each pair.

A Familiar Pair, A New Question

Remember the certified pair, and ?

Instead of full algebra, what if you just tested one number?

Numeric Quick-Check Before Full Verification

Before full symbolic work, test 2 to 3 values:

• Compute and for a chosen value
• One failing test = definitely not inverses, stop right there
• All tests passing = promising, but still needs symbolic proof

Screening a Pair With Four Checks

, proposed

Two values, four checks, all passing — promising, not yet proven.

When a Numeric Test Catches a Fake

, proposed

One failing test is enough — reject immediately. No algebra needed.

The Asymmetry Behind Numeric Screening

• One failing test always disproves the pair, immediately
• All passing tests never proves the pair, no matter how many

Screening eliminates candidates fast; only symbolic work certifies.

Your Turn: Screen Four Pairs

1. ,
2. ,
3. ,
4. ,

Test each numerically. Which need symbolic follow-up?

A Failed Test Isn't a Dead End

Composition isn't just a judge that certifies or rejects. A failed test
also tells you exactly where to look for the mistake.

Same tool, new role: now you're the one whose work gets inspected.

A Proposed Inverse With a Bug

. A student proposes .

The composition doesn't simplify to — something is wrong.

Trace the Error, Correct It

The result was instead of — off by a constant. Checking the swap-and-solve
steps: the sign on the constant term was flipped.

Corrected:

Your Turn: Find This Error

. A proposed inverse gives .

Where did the algebra go wrong? What should actually be?

One Test, Two Faces of Proof

• Composition simplifying to certifies a genuine inverse relationship
• Composition failing to simplify to locates the specific defect

It's the same test either way — only the outcome changes what it tells you.

Solo Commit: Write Your Verdict

Here is a fresh pair. Write your verdict — inverses, or not — and your reasoning.
No discussion yet.

Your Turn: No Framing, No Hints

A pair is presented below. Nothing tells you whether it's genuinely inverse.

Decide: screen first, or go straight to symbolic work? Then find your verdict.

Watch Out For These Four Mistakes

• One direction only: always check BOTH and
• Numeric is enough: numeric checks only screen; symbolic work proves it
• Simplification slips: spot-check with a number if unsure
• Assumed pairs: a pair given together isn't automatically an inverse pair

What Composition Ultimately Proves for You

A composition that simplifies to is a certificate the relationship holds
for every input at once. One that doesn't tells you exactly where to look
for the problem.

Next: how does this inverse relationship look on a graph, with no algebra at all?