Question:

Step 1: Understand the composition requirement. For the composite function $(fog)(x) = f(g(x))$ to be defined, the output of $g(x)$ must lie within the domain of $f$. Since the domain of $f$ is $[-3, 5]$, we need: $$-3 \leq g(x) \leq 5$$ Step 2: Analyze the lower bound condition. We need to check if $g(x) \geq -3$. Since $g(x) = |3x + 4|$ and absolute values are always non-negative: $$g(x) = |3x + 4| \geq 0$$ Since $0 > -3$, the condition $g(x) \geq -3$ is automatically satisfied for all real $x$. Therefore, we only need to focus on the upper bound. Step 3: Apply the upper bound condition. We need $g(x) \leq 5$, which means: $$|3x + 4| \leq 5$$ Step 4: Solve the absolute value inequality. The inequality $|3x + 4| \leq 5$ is equivalent to: $$-5 \leq 3x + 4 \leq 5$$ Step 5: Isolate the variable $x$. Subtract 4 from all parts: $$-5 - 4 \leq 3x \leq 5 - 4$$ $$-9 \leq 3x \leq 1$$ Divide all parts by 3: $$-3 \leq x \leq \frac{1}{3}$$ Step 6: State the final answer. The domain of $(fog)(x)$ is the set of all $x$ values satisfying the above inequality: $$\boxed{\left[-3, \frac{1}{3}\right]}$$ This matches **Option 2** (and Option 3, as they are identical).