Continuous and invertible function

Question

Let $f(x)=\frac{x^7}{15}+9x^3+25x+1$. If I want to prove that it is a surjective function, can I say that, since it is strictly monotone and continuous, then it is surely invertible and so surjective?

Answer 1

Let $y_0 \in \mathbb R$.

Since $f(x) \to \infty$ as $x \to \infty$, there is some number $b$ such that $f(b) >y_0$.

Since $f(x) \to -\infty$ as $x \to -\infty$, there is some number $a<b$ such that $f(a) <y_0$.

The IVT shows that there is $x_0 \in [a,b]$ such that $f(x_0)=y_0$. This gives $f(\mathbb R)= \mathbb R.$