Disproving a function exists

Question

Prove/Disprove: There exists a function $f: \Bbb R \to\Bbb R$ such that $$ \arctan(f(x)) = 2x/(\cos^2(x) + 3) $$

for every x ∈ $\Bbb R$

I know that it's not true because arctan is limited between $\frac{-0.5}{\pi} \leq x \leq \frac{0.5}{\pi}$, but how can you fully prove it without just giving an example? seeing that it is an "exists" proof

Answer 1

I don't understand your request. Take $x>\pi$. Then$$\frac{2x}{\cos^2(x)+3}\geqslant\frac{2x}4=\frac x2>\frac\pi2.$$Therefore, for such a $x$ you cannot possible have$$\arctan\bigl(f(x)\bigr)=\frac{2x}{\cos^2(x)+3},$$since$$(\forall y\in\mathbb{R}):\arctan(y)<\frac\pi2.$$What can possibly be wrong with this proof?

Answer 2

Exactly what you did.

Suppose it exists. Then for $x = 100$ (some big number), $\arctan f(x) = \frac{200}{\cos^2 100 + 3} > \frac{200}{4} = 50 > \max \arctan$, contradiction.