1. **Problem:** Given $(f \circ g)(x) = 2x + 5$ and $g(x) = \frac{x}{x+1}$, find $f(x)$. 2. **Recall:** The composition $(f \circ g)(x) = f(g(x))$. So, $f(g(x)) = 2x + 5$. 3. **Step:** Substitute $g(x)$ into $f$: $$f\left(\frac{x}{x+1}\right) = 2x + 5$$ 4. **Goal:** Express $f$ in terms of its input $u = \frac{x}{x+1}$. We want $f(u) = ?$ 5. **Solve for $x$ in terms of $u$: $$u = \frac{x}{x+1} \implies u(x+1) = x \implies ux + u = x \implies ux - x = -u \implies x(u - 1) = -u$$ $$x = \frac{-u}{u - 1}$$ 6. **Rewrite $(f \circ g)(x)$ in terms of $u$: $$f(u) = 2x + 5 = 2 \cdot \frac{-u}{u - 1} + 5$$ 7. **Simplify:** $$f(u) = \frac{-2u}{u - 1} + 5 = \frac{-2u}{u - 1} + \frac{5(u - 1)}{u - 1} = \frac{-2u + 5u - 5}{u - 1} = \frac{3u - 5}{u - 1}$$ 8. **Final answer:** $$\boxed{f(x) = \frac{3x - 5}{x - 1}}$$