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  $\frac{-1}{2}\leq \frac{x}{x^2+1}\leq \frac{1}{2}$

Write
$$\frac{4x}{9x^2+25}=\frac{4}{9x+\frac{25}{x}}$$ and use AM-GM in two cases:
1) $x>0$:
$$y\leq\frac{4}{2\sqrt{9x\cdot\frac{25}{x}}}=\frac{2}{15}.$$
2) $x<0$: $$y\geq-\frac{4}{2\sqrt{(-9x)\cdot\left(-\frac{25}{x}\right)}}=-\frac{2}{15}.$$
The equality occurs for $9x=\frac{25}{x}$ and since we have a continuous function,
we got the answer:
$$\left[-\frac{2}{15},\frac{2}{15}\right]$$
