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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?

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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 $f(x) = \frac{1}{x}, g(x) = -\frac{1}{x}$.

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