Onto property of the function

Question

How can we prove the onto property of this non-homogeneous function $$f(x)= (|x|^{p-2} + |x|^{q-2}) x \quad \text{for} \quad x \in \mathbb{R} \quad \text{where} \quad p, q >1.$$

Any ideas (without performing any complicative calculations)?

Answer 1

HINT

Since $f(0)=0$ and $f(x) \to \infty$ as $x \to -\infty$, IVT implies $\mathbb{R}^+ \subset \mathrm{Im}(f)$.

Can you do the other side?