If we know range of a function , how can construct range of other function? 
If we know $\frac{-1}{2}\leq \frac{x}{x^2+1}\leq \frac{1}{2}$ how can obtain the range of the function below?
  $$y=\frac{4 x}{9x^2+25}$$

The problem is what's the range with respect to  $\frac{-1}{2}\leq \frac{x}{x^2+1}\leq \frac{1}{2}$

$$y=\frac{4 x}{9x^2+25}\\y=\frac{4 x}{25(\frac{9}{25}x^2+1)}=\\y=\frac4{25}\frac{ x}{((\frac{3}{5}x)^2+1)}$$ now take $u=\frac{3x}{5}$
$$y=\frac4{25}\frac{ x}{((\frac{3}{5}x)^2+1)}=\\\frac4{25}\frac{ \frac{5}{3}u}{((u)^2+1)}=\\\frac{4}{15}\frac{u}{u^2+1}\\$$you know $\frac{-1}{2}\leq \frac{u}{u^2+1}\leq \frac{1}{2}$ so 
$$-\frac{4}{15}\times \frac {1}{2} \leq y=\frac{4}{15}\frac{u}{u^2+1} \leq \frac{4}{15}\times \frac {1}{2}$$
