Range of function $y=x^2 + \frac{4}{x^2+9}$ 
Please don't give the solution, I already got the answer by a different method.
I want to know why the method in the picture is wrong? Why cannot we simple add inequalities like that to get the interval of range and then find the minimum value from that range?
The correct answer to this question is : 4/9 ( minimum value)


I had a similar question long time back. Why does the range of $\sin x +\cos x$ is [-$\sqrt 2$,$\sqrt 2$] and  not [$-2$,$2$] as the range of $\sin x$ and $\cos x$ is [-1,1], where $x$ is real? Basically why can't you add the ranges of two functions to obtain the overall range of their sum (in general). The catch here is that,the two extreme points of their range do not occur for the same value of $x$,in general. Hence you can't add their ranges, as there will be no such $x$ where that value occurs. In your case you can't add the ranges of the functions $t$ and $4/t$ as the minima/maxima for these functions occur for different values of $t$.
