Question

Determine the values of n for which f(x)=x" has an inverse that is a function. Assume that n is a whole number.

Answer 1

Assuming you're asking "for which values of $n$ the function $x^n$ has an inverse that is a function", the answer is "all the odd exponents $n$".

Infact, if $n$ is even, you have that

$x^n=(-x)^n \quad \forall x \in \mathbb{R}$

and so $f(x)=x^n$ is not injective, and thus not invertible

On the other hand, if $n$ is odd, we have:

• $\lim_{x\to\pm\infty}x^n=\pm\infty$
• $x^n$ is continuous.
• The first two points tell us that the function is surjective.
• Moreover, the derivative is $f'(x)=nx^{n-1}$. Since $n-1$ is even, we have $f'(x)>0$, thus the function is always increasing, and so the function is also injective.
• Injective and surjective means bijective, and the function can be inverted.