Ensuring Well-Defined And Correct Behavior: Key Domains To Check When Performing Operations On Functions

domains that need to be checked when adding/subtracting/multiplying/dividing f(x) and g(x) for h(x)

domain of inputs f(x) and g(x), and final result h(x)

When adding, subtracting, multiplying, or dividing two functions f(x) and g(x) to obtain a new function h(x), there are several domains that need to be checked to ensure that the resulting function is well-defined and behaves correctly in all possible situations. These domains include:

1. Domain of f(x) and g(x): It is important to check that f(x) and g(x) have the same domain. This ensures that each expression in the operation makes sense and can be evaluated for all values of x in the domain.

2. Domain of the resulting function h(x): After the operation is performed, it is important to check that the resulting function h(x) has a well-defined domain. In some cases, the domain of h(x) may be different from the domains of f(x) and g(x). For example, if f(x) = sqrt(x) and g(x) = 1/x, then h(x) = f(x) + g(x) = sqrt(x) + 1/x has a domain of x > 0.

3. Division by zero: When dividing two functions, it is important to check that the denominator g(x) is not zero for any value of x in the domain. If there are any values of x for which g(x) = 0, the resulting function will be undefined at those points.

4. Common factors: When adding or subtracting two functions, it is important to check for common factors and simplify the expression if possible. For example, if f(x) = x^2 – 4 and g(x) = x – 2, then h(x) = f(x) + g(x) = (x + 2)(x – 2) + (x – 2) = (x + 2)(x – 1).

5. Composite functions: When performing operations on composite functions (functions of functions), it is important to check that the functions can be composed in the correct order. For example, if f(x) = sqrt(x) and g(x) = x^2, then h(x) = f(g(x)) = sqrt(x^2) has a domain of x >= 0, since the square root function is only defined for non-negative inputs.

Overall, checking these domains and simplifying the expression as much as possible will help ensure that the resulting function is well-defined and behaves as expected.

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