Math · College algebra · Worked example
Restrict a domain to find an inverse
f(x) = (x − 2)² is not one-to-one. Restrict its domain so that it has an inverse, and find that inverse.
See why there is no inverse yet
f(1) = f(3) = 1: two inputs give the same output, so the horizontal line y = 1 crosses the graph twice and no single inverse can send 1 back.
Restrict the domain
Keep the right half of the parabola, x ≥ 2. On it, f is increasing, so it is one-to-one, and its outputs are y ≥ 0.
Solve for x
y = (x − 2)² gives x − 2 = √y, taking the positive root because x ≥ 2. So x = 2 + √y.
Swap and state the domain
f⁻¹(x) = 2 + √x, for x ≥ 0: the range of the restricted f becomes the domain of the inverse. Check: f(5) = 9 and 2 + √9 = 5.
Result
On x ≥ 2, f⁻¹(x) = 2 + √x, for x ≥ 0.
Your turn
On the domain x ≥ 0, find the inverse of f(x) = x² + 1.
Show the answer and explanation
f⁻¹(x) = √(x − 1), for x ≥ 1.
For x ≥ 0 the outputs are y ≥ 1. Solve y = x² + 1: x = √(y − 1), the nonnegative root. Swap: f⁻¹(x) = √(x − 1) for x ≥ 1. Check: f(2) = 5 and √(5 − 1) = 2.
Keep exploring
Restrict to the left half, x ≤ 2, instead. Then the inverse takes the negative root: f⁻¹(x) = 2 − √x, which sends 9 back to −1.
Return to the concept →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
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