Question:

Step 1: Set up the composition of functions. We need to find $(fog)(x) = f(g(x))$. First, let's identify what $g(x)$ equals, then substitute it into $f$. $$g(x) = \frac{3x}{4} - \frac{|x|}{4}$$ So we need to evaluate: $$f(g(x)) = f\left(\frac{3x}{4} - \frac{|x|}{4}\right)$$ Step 2: Simplify $g(x)$ for the case when $x \geq 0$. When $x \geq 0$, we have $|x| = x$. Therefore: $$g(x) = \frac{3x}{4} - \frac{x}{4} = \frac{3x - x}{4} = \frac{2x}{4} = \frac{x}{2}$$ Step 3: Apply $f$ to $g(x)$ when $x \geq 0$. Now we substitute $u = \frac{x}{2}$ into $f(u) = 3u + |u|$: Since $x \geq 0$, we have $u = \frac{x}{2} \geq 0$, so $|u| = u = \frac{x}{2}$. $$f(g(x)) = f\left(\frac{x}{2}\right) = 3 \cdot \frac{x}{2} + \frac{x}{2} = \frac{3x}{2} + \frac{x}{2} = \frac{4x}{2} = 2x$$ Step 4: Simplify $g(x)$ for the case when $x < 0$. When $x < 0$, we have $|x| = -x$. Therefore: $$g(x) = \frac{3x}{4} - \frac{(-x)}{4} = \frac{3x}{4} + \frac{x}{4} = \frac{3x + x}{4} = \frac{4x}{4} = x$$ Step 5: Apply $f$ to $g(x)$ when $x < 0$. Now we substitute $u = x$ into $f(u) = 3u + |u|$: Since $x < 0$, we have $|u| = |x| = -x$. $$f(g(x)) = f(x) = 3x + |x| = 3x + (-x) = 3x - x = 2x$$ Step 6: Conclude the result. In both cases (whether $x \geq 0$ or $x < 0$), we obtain: $$(fog)(x) = 2x$$ Therefore, the answer is **Option 4: $(fog)(x) = 2x$** <div class="key-concept"><strong>Key Concept:</strong> Composition of functions involving absolute value</div> <div class="trap-box"><strong>Trap:</strong> Not considering cases for positive and negative x separately when dealing with absolute value functions.</div>