Injectivity and inverse of x³+1

Problem

Establish whether $f(x) = x^3 + 1$ is injective, surjective, bijective on $\mathbb{R}\to\mathbb{R}$, and compute its inverse.

Solution

Injective: $x^3$ is strictly increasing on all of $\mathbb{R}$, so $x^3+1$ is too; a strictly monotonic function is injective. ✓ Surjective: as $x$ ranges over $\mathbb{R}$, $x^3$ takes all real values, hence $x^3+1$ covers all of $\mathbb{R}$: $\mathrm{Im}(f)=\mathbb{R}$. ✓ Bijective: being injective and surjective, yes. ✓

Inverse: set $y=x^3+1$, isolate $x$: $x^3=y-1$, hence $x=\sqrt[3]{y-1}$. Swapping the names of the variables: $\boxed{f^{-1}(x)=\sqrt[3]{x-1}}.$

Links

Topics: Functions and properties
Concepts: Bijective function · Injective function · Inverse function · Surjective function
Functions: Cubic function
Skills: Solving equations · Studying a function
Exercise types: Study of a function