Find the condition for $f^{-1}$ to exist.

Question

If $f: \mathbb R \to \mathbb R$ defined by $f(x)= x^3+ px^2 + qx+ k \sin{x}$, where $k,p,q \in \mathbb R$.

Find the condition for $f^{-1}$ to exist.

Can somebody please give me a Hint how to solve this problem.

EDIT(After getting hints): If we can show that $f$ is strictly increasing, then we can get that $f$ is injective. So we want $f^{'}(x)>0$ and that gives me $p^2 < 3(q + k \cos{x}).$

As $f^{'}(x)$ will be a quadratic polynomial and we want it to be strictly positive or strictly negative but as coefficient of $x^2$ is positive, we can only get derivative to be strictly positive if discriminant is $<0.$

Answer 1

Guide:

Check $\lim_{x \rightarrow \infty} f(x)$.

Check $\lim_{x \rightarrow -\infty} f(x)$.

Note that $f$ is continuous.

Find conditions to make it strictly increasing.

Answer 2

Strictly increasig functions are injective, and hence, have inverse.