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): It is stated that the domain of $(f + g)(x) = e^x + \log x$ is $\mathbb{R}$. This is incorrect. The function $f(x) = e^x$ is defined for all $x \in \mathbb{R}$, but $g(x) = \log x$ is only defined for $x>0$. Consequently, the domain of $(f + g)(x)$ is $(0, \infty)$, not $\mathbb{R}$. Thus, Assertion (A) is false.
- Reason (R): The reason states that the domain of the sum of two functions is the intersection of their individual domains. This is correct. The domain of $f(x) = e^x$ is $\mathbb{R}$, and the domain of $g(x) = \log x$ is $(0, \infty)$. The intersection of these domains is $(0, \infty)$. Therefore, Assertion (A) is false, but Reason (R) is true.