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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$\begingroup$ Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. $\endgroup$– José Carlos SantosMay 18, 2021 at 7:35
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1$\begingroup$ Can you clarify what why we can find the domain of a function by taking the intersection of domains of individual functions means? At least by providing an example! $\endgroup$– mathcounterexamples.netMay 18, 2021 at 7:35
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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 $f(x) = \frac{1}{x}, g(x) = -\frac{1}{x}$.
