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Restricted Function

Restricted Function

A function with a restricted domain.

Worked Example

Problem: Restrict the function f(x) = x² so that it becomes one-to-one, then find its inverse.
Step 1: The natural domain of f(x) = x² is all real numbers, but the function is not one-to-one because, for example, f(3) = f(−3) = 9.
f(x)=x2,x∈Rf(x) = x^2, \quad x \in \mathbb{R}
Step 2: Restrict the domain to x ≥ 0. The restricted function is now:
g(x)=x2,x≥0g(x) = x^2, \quad x \geq 0
Step 3: On this restricted domain, g is one-to-one (each output corresponds to exactly one input), so its inverse exists.
g−1(x)=x,x≥0g^{-1}(x) = \sqrt{x}, \quad x \geq 0
Answer: The restricted function g(x) = x² with x ≥ 0 is one-to-one and has the inverse g⁻¹(x) = √x.

Why It Matters

Restricting a function is essential when you need to find an inverse. Many common functions—like x2x^2, sin⁡x\sin x, and cos⁡x\cos x—are not one-to-one on their full domains, so their inverses (x\sqrt{x}, arcsin⁡x\arcsin x, arccos⁡x\arccos x) are defined only by first restricting the original function to a suitable interval. Without restriction, these inverse functions would not exist.

Common Mistakes

Mistake: Thinking a restricted function is a different function entirely, unrelated to the original.
Correction: A restricted function uses the same rule as the original; only the set of allowed inputs changes. The formula stays the same—the domain shrinks.

Related Terms