AP® Precalculus · Unit 2 · Topic 2.8
Inverses as a Reversed Mapping
An inverse function undoes what the original function did, by swapping every input with its output. If f sends a to b, f−1 sends b back to a.
A function is a set of arrows: each input points to one output. If f(2) = 5, that arrow goes from 2 to 5. The inverse function, written f−1, just flips every arrow around: it takes 5 back to 2. That is the whole idea. Nothing about the rule changes except the direction you read it.
You can see this directly in a table. If f pairs (1, 4), (2, 7), (3, 10), then f−1 pairs (4, 1), (7, 2), (10, 3) — same pairs, columns swapped. For a formula, the swap-and-solve method does the same job algebraically: write y = f(x), swap x and y, then solve for y. That solved expression is f−1(x). Because the pairing is exactly reversed, composing the two functions in either order cancels the trip: f(f−1(x)) = x and f−1(f(x)) = x. That check is how you confirm an inverse is right, not just plausible.
This reversed-mapping idea is what makes logarithms make sense as the inverse of exponential functions: logb(x) exists precisely because it swaps the input-output pairs of bx. Everything coming next in logarithms — their graphs, their domain and range, their algebra rules — traces back to this one fact about reversing arrows. Keep the table picture in mind and the logarithm rules stop feeling arbitrary.
Key terms
- Inverse function
- A function that reverses every input-output pair of the original function.
- Swap-and-solve
- The method of exchanging x and y in y = f(x), then solving for y, to find f−1(x).
- Composition check
- Verifying f(f−1(x)) = x and f−1(f(x)) = x to confirm two functions are true inverses.
- Reciprocal of a function
- The expression 1/f(x), formed by dividing 1 by the output, unrelated to the inverse function.
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End of topic.