Lesson 9 of 12

Follow membership through a function

Compare inverse images, which preserve intersections, with images, which can merge distinct inputs.

Loading your saved responses…

Try

A function can send distinct inputs to the same output. That merging explains why tracing outputs backward behaves differently from sending sets forward. Neither operation requires guessing an inverse formula.

Predict

Predict the inverse image of an output set under squaring. Both positive and negative inputs may belong. For f:X→Y and B⊆Y, f⁻¹(B) is the set of x in X whose output belongs to B. This notation does not assert that f has an inverse function.

Experiment

Draw arrows for inputs −1,0,1 under squaring. Circle output 1 and collect every input arrow reaching it. Next split the input domain into the singleton sets containing −1 and 1. Their images coincide even though the input sets are disjoint. Record precisely where a shared output has two different witnesses.

Understand

Worked argument: inverse images preserve unions. For any x∈X, membership in f⁻¹(B∪C) means f(x)∈B or f(x)∈C. That is equivalent to x∈f⁻¹(B) or x∈f⁻¹(C), hence membership in their union. Every step is reversible, proving equality.

An invalid argument for images preserving intersections says an output in both images must come from one input in both sets. It may instead come from two distinct inputs. The arrow diagram exposes the unsupported identification.

Apply

Use an arbitrary input to prove an inverse-image identity. For an image claim, keep track of existential input witnesses separately until you have a reason they are equal. Specify domains and codomains so all sets and function applications are meaningful.

Continue

Depth: images always satisfy f(A∩D)⊆f(A)∩f(D). Prove this inclusion, then show that injectivity allows the reverse inclusion by forcing the two witnesses to agree. This explains why the missing hypothesis is structural rather than a numerical accident.

Apply · explain · revise

Practice and reasoning

Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.

Let f(x)=x² on domain {−2,−1,0,1,2}. Enter the inverse image of {1,4}.

Your solution set

Integers, fractions and finite decimals are accepted. Separate values with commas; type none for no solutions. Expressions such as sqrt(2) are not parsed by this checker.

Must a function be injective to have an inverse image of a set?

Your responseYesNo

For f:X→Y and subsets B,C of Y, prove f⁻¹(B∩C)=f⁻¹(B)∩f⁻¹(C).

Your argument or explanation

Structure your argument and preserve revisions

These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.

Claim or conclusionAssumptions, domains and exceptional casesDefinitions and theorems usedChecks, counterexamples and unresolved gaps

Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.

Give a two-input function and sets A,D for which f(A∩D) is not f(A)∩f(D). Explain the failure.

Your argument or explanation

Structure your argument and preserve revisions

Claim or conclusionAssumptions, domains and exceptional casesDefinitions and theorems usedChecks, counterexamples and unresolved gaps

Reflect and continue

Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.

Next: Which properties survive composition? ·Try the mixed unit assessment ·Unit overview · Saved mathematics and backups