Question

In a physics class, the teacher proposes that the velocity of a car in miles per hour and the stopping distance in feet are represented by inverse functions.

Assuming this is true, which graph includes a pair of functions for a specific velocity function and the corresponding stopping distance that can be verified as inverses?

Answer 1

You should pick the graph that shows two curves which are mirror images across the line y = x (one curve a parabola-like shape and the other its square-root reflection). Inverse functions reflect across y = x, so the correct graph is the one where every point (v, d) on the "velocity→distance" curve corresponds to a point (d, v) on the other curve.

Example to make this concrete:
- Let f(v) = 0.04 v^2 (v in mph → stopping distance in ft).
- Its inverse is f^{-1}(d) = sqrt(d/0.04) = 5 sqrt(d) (d in ft → v in mph).

Verify: f^{-1}(f(v)) = 5 sqrt(0.04 v^2) = 5·0.2·v = v, so they are inverses. On the graph, f is an upward-opening parabola (for v ≥ 0) and f^{-1} is a square-root curve; they are symmetric about y = x. Pick the graph that shows exactly that symmetry.