$f(x) = \sqrt{ |x|} - \log (1+|x|)$ has no Upper Bound on $\mathbb R$ Is the following  function bounded on  $\mathbb R$ ?

$f(x) = \sqrt {|x|} - \log (1+|x|)$ 

My Attempt:  I can prove that $f(x) $ is an increasing function on $(0 , \infty)$ and $f(x) $ is decreasing on $(-\infty ,0)$ and $f(x) $ is continuous on $\mathbb R$ and $f(x) \geq 0$  for all $x \in  \mathbb R$. But I can not proceed further.

Can anyone please give me any hint or some thing?
Let $ x = e^{k}-1$ where $ k &gt;&gt; 2$.   

Then $ \sqrt{x} &gt; e^{k/2} - 1$ and $ \log (1 + x) = k$, and clearly $e^{k/2} - k  - 1 $ tends to infinity.
Hint: use (prove) the fact that for any $M &gt; 0$ there is $X &gt; 0$ that for all $x &gt; X$

$$ x^\frac{1}{2} &gt; M + log(1 + x)$$