Question

Assertion (A): Let $f(x) = e^x$ and $g(x) = \log x$. Then $(f + g)(x) = e^x + \log x$ where the domain of $(f + g)$ is $\mathbb{R}$.
Reason (R): $\text{Dom}(f + g) = \text{Dom}(f) \cap \text{Dom}(g)$.

• A. Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

When working with the sum of functions, always consider the intersection of their individual domains to determine the domain of the sum.

Answer 1

The Correct Option is D

- Assertion (A): The assertion claims that the domain of $(f + g)(x) = e^x + \log x$ is $\mathbb{R}$.
However, this is incorrect. While $f(x) = e^x$ is defined for all $x \in \mathbb{R}$, the function $g(x) = \log x$ is only defined for $x > 0$. 
Therefore, the domain of $(f + g)(x)$ is not $\mathbb{R}$, but rather $(0, \infty)$. 
So, Assertion (A) is incorrect. 
- Reason (R): The reason is correct. The domain of the sum of two functions is the intersection of the domains of the individual functions. 
The domain of $f(x) = e^x$ is $\mathbb{R}$, and the domain of $g(x) = \log x$ is $(0, \infty)$. Thus, the domain of $(f + g)(x)$ is the intersection of these two domains, which is $(0, \infty)$. Thus, Assertion (A) is incorrect, but Reason (R) is correct.