Question

If $f(x) = \sqrt{2 - x^2}$ and $g(x) = \log(1 - x)$ are two real-valued functions, then the domain of the function $(f+g)(x)$ is:

• A. $[-2, 2]$
• B. $[-2, 1)$
• C. $(-\infty, 1)$
• D. $(1, 2]$

Hint:

When adding functions, the domain is the intersection of individual domains.

Answer 1

The Correct Option is B

Domain of $f(x) = \sqrt{2 - x^2}$ requires $2 - x^2 \geq 0 \Rightarrow x \in [-\sqrt{2}, \sqrt{2}]$
Domain of $g(x) = \log(1 - x)$ requires $1 - x>0 \Rightarrow x<1$
Thus, domain of $(f + g)(x) = f(x) + g(x)$ is the intersection: $[-\sqrt{2}, \sqrt{2}] \cap (-\infty, 1) = [-\sqrt{2}, 1)$
But $[-\sqrt{2}, 1) \subset [-2, 1)$ ⇒ boxed domain: $[-2, 1)$