<s>[INST] 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  $\frac{-1}{2}\leq \frac{x}{x^2+1}\leq \frac{1}{2}$
 [/INST] We have
\begin{eqnarray*}
- \frac{1}{2} \leq \frac{x}{x^2+1} \leq \frac{1}{2}
\end{eqnarray*}
Let $x= \frac{3z}{5}$
\begin{eqnarray*}
- \frac{1}{2} \leq \frac{\frac{3z}{5}}{(\frac{3z}{5})^2+1} \leq \frac{1}{2} \\
- \frac{1}{2} \leq \frac{15z}{9z^2+25} \leq \frac{1}{2}
\end{eqnarray*}
Now multiply by $  \frac{4}{15}$
\begin{eqnarray*}
- \frac{2}{15} \leq \frac{4z}{9z^2+25} \leq \frac{2}{15}.
\end{eqnarray*}
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