Range of sum of two functions.

Question

How to find range of $\tan^{-1}x + 10$

My attempt : I thinks that range of sum of two functions $f$ and $g$ is either range of $f$ + range of $g$ or Union of range of $f$ and range of $g$. Please help me.

Answer 1

Note

$$-\frac\pi2<\tan^{-1}x<\frac\pi2$$

Then,

$$-\frac\pi2+10<\tan^{-1}x+10<\frac\pi2 +10$$

Thus, the range is $(-\frac\pi2+10,\frac\pi2 +10)$.

Answer 2

Suppose you have $f(x)= (x+1)^2, g(x) = (x-1)^2$ then the range is of each is $[0,\infty)$ But the sum of the two...

$f(x)+g(x) = (x+1)^2 + (x-1)^2 = 2x^2 + 2$

The range is $[2,\infty)$

The range of the sum of the two functions may not be obvious in looking at the ranges of each.