Hello, Eraser147!
$\displaystyle \text{Find the range of: }\:f(x) \:=\:\frac{x+2}{x^2+5}$
An informal sketch gives us a serpentine curve.
The derivative simplifies to: .$\displaystyle f'(x) \;=\;-\frac{(x-1)(x+5)}{(x^2+5)^2}$Code:| | * | * : * * : * -5 *| : * -----------+------*-+----+------------- * : * | 1 * : * | * : * | * | |
Hence, the extrema are at: $\displaystyle x = 1,\text{-}5$
And: .$\displaystyle \begin{Bmatrix}f(1) &=& \frac{1}{2} \\ \\[-4mm] f(\text{-}5) &=& \text{-}\frac{1}{10} \end{Bmatrix}$
Therefore, the range is: $\displaystyle \left[\text{-}\tfrac{1}{10},\:\tfrac{1}{2}\right]$