Suppose f1(x) and f2(x) are functions defined on domains D1 subset R and D2 subset R respectively and codomains are subset of R. What will be the domain of the function f * 1(x) + f * 2(x)?

General
1

Suppose f1(x) and f2(x) are functions defined on domains D1 subset R and D2 subset R respectively and codomains are subset of R. What will be the domain of the function f * 1(x) + f * 2(x) ?

2

Suppose f_1(x) and f_2(x) are functions defined on domains D_1 \subseteq \mathbb{R} and D_2 \subseteq \mathbb{R} respectively, and codomains are subsets of \mathbb{R}. What will be the domain of the function f_1(x) + f_2(x)?

Answer:

To determine the domain of the function f_1(x) + f_2(x), we need to consider the domains of the individual functions f_1(x) and f_2(x). The domain of a function is the set of all possible inputs (values of x) for which the function is defined.

Given:

  • f_1(x) is defined on domain D_1 \subseteq \mathbb{R}.
  • f_2(x) is defined on domain D_2 \subseteq \mathbb{R}.

For the function f_1(x) + f_2(x) to be defined at a particular value of x, both f_1(x) and f_2(x) must be defined at that value of x. This means that x must be in both D_1 and D_2 simultaneously.

Thus, the domain of the function f_1(x) + f_2(x) is the intersection of the domains of f_1(x) and f_2(x).

Mathematically:

\text{Domain of } (f_1(x) + f_2(x)) = D_1 \cap D_2

Explanation:

  • The intersection D_1 \cap D_2 represents all values of x that are common to both D_1 and D_2.
  • If x \in D_1 \cap D_2, then x is in both D_1 and D_2, meaning both f_1(x) and f_2(x) are defined at x.
  • Therefore, f_1(x) + f_2(x) is also defined at x.

Example:

Suppose:

  • f_1(x) is defined on D_1 = [1, 5].
  • f_2(x) is defined on D_2 = [3, 7].

The domain of f_1(x) + f_2(x) would be:

D_1 \cap D_2 = [1, 5] \cap [3, 7] = [3, 5]

So, the domain of f_1(x) + f_2(x) is [3, 5].

In conclusion, the domain of the function f_1(x) + f_2(x) is the intersection of the domains of f_1(x) and f_2(x).

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