1. **State the problem:** We are given two functions $f(x) = 3x - 4$ and $g(x) = 4x + 1$. We need to find: (a) $f(-2)$ (b) The inverse function $f^{-1}(x)$ (c) The composition $fg(x)$ and find constants $a$ and $b$ such that $fg(x) = ax + b$ 2. **Find $f(-2)$:** Substitute $x = -2$ into $f(x)$: $$f(-2) = 3(-2) - 4 = -6 - 4 = -10$$ 3. **Find the inverse function $f^{-1}(x)$:** The inverse function reverses the roles of $x$ and $y$. Start with: $$y = 3x - 4$$ Swap $x$ and $y$: $$x = 3y - 4$$ Solve for $y$: $$3y = x + 4$$ $$y = \frac{x + 4}{3}$$ So, $$f^{-1}(x) = \frac{x + 4}{3}$$ 4. **Find $fg(x)$:** The composition $fg(x)$ means $f(g(x))$. Substitute $g(x)$ into $f$: $$fg(x) = f(g(x)) = f(4x + 1) = 3(4x + 1) - 4$$ Simplify: $$= 12x + 3 - 4 = 12x - 1$$ 5. **Identify $a$ and $b$:** From $fg(x) = ax + b$, we have: $$a = 12, \quad b = -1$$ **Final answers:** (a) $f(-2) = -10$ (b) $f^{-1}(x) = \frac{x + 4}{3}$ (c) $a = 12$, $b = -1$