Showing a function is inversible

Question

Hi I'm struggling with this question.

Let $f : ℝ → ℝ$ be defined by $f(x)=\frac{5}{\sqrt {x^3+4}}$.

Show that this function is invertable.

Answer 1

It is clear that $f(-x)=\frac{|2|}{\sqrt{(-x)^2+1}}=\frac{|2|}{\sqrt{(x)^2+1}}=f(x)$, $\forall x\in \mathbb{R}$; hence $f$ cannot be injective (because opposite numbers gets the same image). On the other hand, $f(x)\ne 0$ $\forall x\in \mathbb{R}$, and because of that $f$ cannot be surjective (in fact, $f(x)> 0$, $\forall x$).

Answer 2

We see $$f(\pm 1) = \frac{2}{\sqrt{(\pm 1)^2 +1}} = \frac{2}{\sqrt 2} =\sqrt{2}.$$ Thus $f(1) = f(-1)$ but $1 \neq -1$ so $f$ is not injective. Also, $f$ is not surjective since for example, there is no $x$ such that $f(x) = 0$.