Inverse Functions & Proofs

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

If f : X → Y is bijective, then it has an inverse function f⁻¹ : Y → X such that:

f⁻¹(f(x)) = x for all x ∈ X,
f(f⁻¹(y)) = y for all y ∈ Y.

To prove a function g is the inverse of f, you must show these two compositions act like identities.

Often, proving a function has an inverse boils down to:

Proving f is bijective.
Explicitly constructing the inverse rule and verifying the compositions.

Example: f : ℝ → ℝ, f(x) = 3x + 1.

Solve y = 3x + 1 ⇒ x = (y − 1)/3.
Define f⁻¹(y) = (y − 1)/3.
Then verify f⁻¹(f(x)) = x and f(f⁻¹(y)) = y.

TL;DR — key idea

Inverse functions exist exactly for bijections; to prove a function has an inverse, you show it is bijective and verify the two-sided inverse identities.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

1
Let f : ℝ → ℝ be defined by f(x) = 5x − 2. 1. Show that f is bijective. 2. Find an explicit formula for f⁻¹. 3. Verify both compositions f⁻¹(f(x)) and f(f⁻¹(y)) give back the input.

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.