Mathematics

Mathematics

Problem

Explain, in one or two sentences, why the function f:R→Rf:\mathbb{R}\to\mathbb{R} defined by f(x)=x2f(x)=x^{2} has no inverse on R\mathbb{R}, but does have an inverse when its domain is restricted to [0,∞)[0,\infty).
Intuitive explanation

A function needs to be one-to-one (injective) to have an inverse, and f(x)=x2f(x) = x^2 fails this on all of R\mathbb{R} because both 22 and −2-2 map to 44. Restricting to [0,∞)[0, \infty) eliminates this duplication—each output comes from exactly one input—making the function bijective and therefore invertible.

Intuitive explanation

A function needs to be one-to-one (injective) to have an inverse, and f(x)=x2f(x) = x^2 fails this on all of R\mathbb{R} because both 22 and −2-2 map to 44. Restricting to [0,∞)[0, \infty) eliminates this duplication—each output comes from exactly one input—making the function bijective and therefore invertible.

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