Mathematics

To show a function is injective, you need to prove that different inputs always produce different outputs—or equivalently, that whenever $f(a) = f(b)$, you must have $a = b$. For $f(x) = x^3$, the key insight is that the cube function is strictly monotonic: it's always increasing, which means it can never "double back" and hit the same output twice. Start by assuming $f(a) = f(b)$ and use algebra to show this forces $a = b$.

To show a function is injective, you need to prove that different inputs always produce different outputs—or equivalently, that whenever $f(a) = f(b)$, you must have $a = b$. For $f(x) = x^3$, the key insight is that the cube function is strictly monotonic: it's always increasing, which means it can never "double back" and hit the same output twice. Start by assuming $f(a) = f(b)$ and use algebra to show this forces $a = b$.