# why we can find domain of a function by taking intersection of domains of individual functions but not range

why we can find domain of a function by taking intersection of domains of individual functions but not range by taking intersection of ranges of individual function. for example- for finding range of f(x) = cos^(-1)x + cot^(-1)x we will domain by intersection of domains of cos inverse and cot inverse and then find range by putting them into equation to find maxima and minima? why?

I assume you mean that $$\arccos x$$ has a range of $$[0, \pi]$$, and $$\text{arccot } x$$ also has a range of $$[0, \pi]$$, but the range of $$\arccos x + \text{arccot } x$$ is not $$[0, 2 \pi]$$ as you have mentioned.
This is because there is no value of $$x$$ such that $$\arccos x = 0$$ and $$\text{arccot } x = 0$$. However, if you have $$f(x) + g(x)$$ and $$f(x)$$ is not defined at $$x = 0$$, $$f(x) + g(x)$$ is not defined unless the term with the discontinuity at $$x = 0$$ disappears completely ($$\frac{1}{x}, \ln x$$ etc.) For example, take $$f(x) = \frac{1}{x}, g(x) = -\frac{1}{x}$$.