Answer
The range is [2,3)
Work Step by Step
This graph will have a minimum if the numerator (and therefore the whole fraction) is zero. That is the case for x =0
So $f(0) = 2 + \frac{0^2}{0^2 +4}$
So $f(0) = 2 + \frac{0}{4}$
So $f(0) = 2 + 0$
So $f(0) = 2$
This graph will have a maximum if the fraction is as big as possible. That is the case if the denominator is the biggest. Since $x^2$ can be increased infinitely, this is an asymptote and it can be calculated by taking a very big value for x.
So $f(100) = 2 + \frac{100^2}{100^2 +4}$
$f(100) = 2 + \frac{10000}{10000 + 4}$
$f(100) = 2 + \frac{10000}{10004}$
$f(100) \approx 2 + 0.9996$
$f(100) \approx 2.9996$
So the asymptote will be y=3
Therefore the range for this function will be [2,3).
