Real analysis: is there a simple way to get an upper and lower limit for the function $ f (x) = frac { sin ( sin {(x)} + x)} {2+ cos {( lvert x rvert} + cos {(x)})} $

Leave $ f: mathbb {R} rightarrow mathbb {R} $

$$ f (x) = frac { sin ( sin {(x)} + x)} {2+ cos {( lvert x rvert} + cos {(x)})} $$

I want to check the lower and upper limits. Here is what I was thinking:

The image / range of $ without {(x)} $ Y $ cos {(x)} $ is $[-1,1]$. Therefore, assuming $ without {x} = 1 $ Y $ cos {x} = – 1 $ I get: $$ f_ {max} = frac {1} {2-1} = 1 $$

The other possibility is $ without {x} = – 1 $ Y $ cos {x} = – 1 $:

$$ f_ {min} = frac {-1} {2-1} = – 1 $$

So it seems to me that $ f (x) $ I would never have values ‚Äč‚Äčoutside $[-1,1]$. However, while that interval could be a lower and upper limit, I would like to be a little more specific and determine the maximum / minimum or the minimum / maximum of $ f (x) $. Any idea how I would do that?