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If f1 x an...

Question

If $f_{1} (x)$ and $f_{_{2}} (x)$ are defined on domains $D_{1}$ and $D_{2}$ respectively, then $f_{1} (x) + f_{2} (x)$ is defined on $D_{1} \cup D_{2} .$ True or false?
If true enter 1 else 0.

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Solution

If $f_{1} (x)$ is defined in the domain of $x ϵ D_{1}$ and $f_{2} (x)$ is defined in the domain of $x ϵ D_{2}$ then $f_{1} (x) + f_{2} (x)$ is defined in the domain $x ϵ D_{1} \cap D_{2}$ .
Consider the possibility that it is defined in the domain $D_{1} \cup D_{2}$
Then for any element in $D_{1}$ , $f_{2} (x)$ won't be defined. Similarly for any element in $D_{2}$ , $f_{1} (x)$ won't be defined. Hence the above statement is false.
Thus the correct answer is $D_{1} \cap D_{2}$

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