Choosing an inverse branch

A quadratic on the whole real line is usually not one-to-one: two inputs give the same output. Restricting to one side of its turning point can make an inverse possible. When rearrangement produces a positive and a negative square root, the original domain selects the permitted branch. Keep that restriction throughout the calculation.

The inverse takes inputs from the original range and produces outputs in the original domain. Verify the formula by composing in both orders on those sets. The square root requires a nonnegative radicand. Remember that √((x − h)²) is |x − h|; replacing it with x − h without checking its sign can select the wrong branch.

Worked example

Find the inverse of f(x) = (x − 2)² + 3 on the domain x ≤ 2. State the inverse domain and range.

1. Write y − 3 = (x − 2)². Since x − 2 ≤ 0, x − 2 = −√(y − 3).
2. Rearrange x = 2 − √(y − 3), then exchange input and output letters.
3. The original range is y ≥ 3 and the original domain is x ≤ 2.

Answer: f⁻¹(x) = 2 − √(x − 3), with domain x ≥ 3 and range f⁻¹(x) ≤ 2.

Revise first: Inverse functions, Completing the square.