If f(x) and g(x) are two functions, then which of the following operations may increase the domain of the resulting function?a)f(x) + g(x)b)f(x) – g(x)c)Both (1) and (2)d)Neither (1) nor (2)Correct answer is option 'C'. Can you explain this answer?

Question

If f(x) and g(x) are two functions, then which of the following operations may increase the domain of the resulting function?a)f(x) + g(x)b)f(x) &ndash; g(x)c)Both (1) and (2)d)Neither (1) nor (2)Correct answer is option 'C'. Can you explain this answer?

Answer

Explanation:

Domain of a Function:
The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Operations on Functions:

Addition (f(x) + g(x)): When two functions f(x) and g(x) are added, the resulting function may have a broader domain than either of the original functions. This is because the sum of two functions can be defined for all values where both individual functions are defined.
Subtraction (f(x) - g(x)): Similarly, when two functions f(x) and g(x) are subtracted, the resulting function may also have a broader domain than the individual functions. However, it is important to note that the domain of the resulting function is determined by the intersection of the domains of the original functions.

Conclusion:
Both addition and subtraction of functions can potentially increase the domain of the resulting function. Therefore, the correct answer is option C - Both (1) and (2).

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