Precalculus A — Functions & Transformations

Inverse functions

continues from lesson 3 — values defined earlier in the course stay live here

ELI5: an inverse undoes the original — if f turns 5 into 16, then f⁻¹ turns 16 back into 5. Only one-to-one machines have inverses (each output from exactly one input), which is why we restrict x ≥ 0 before inverting x².

sq\left(x\right) = x^{2}

restricted to x ≥ 0

sq_{inv}\left(y\right) = \sqrt{y}

its inverse

✓ pass

\mathrm{abs}\left(\mathrm{sq_{inv}}\left(\mathrm{sq}\left(7\right)\right) - 7\right) < 10^{-9}

undo really undoes: √(7²) = 7

Try it yourself: the inverse of f(x) = 3x + 1 is (y − 1)/3. Define finv(y) so that it undoes f.

finv\left(y\right) = \frac{y - 1}{3}

✏️ Your turn: define finv(y) = the inverse of 3x + 1. Green when finv undoes f at x = 5 and x = −2.

✓ pass

abs(finv(3 * 5 + 1) - 5) < 1e-9 and abs(finv(3 * (-2) + 1) + 2) < 1e-9

green when finv really reverses f

Where next: Precalculus B studies the function families calculus leans on — polynomials and rational functions — and previews the limit.