1. **Problem Statement:** Find the inverse of each function given. 2. **Formula and Rules:** To find the inverse function $f^{-1}(x)$, swap $x$ and $y$ in the equation $y=f(x)$ and solve for $y$. 3. **Step-by-step Solutions:** **a) $f(x) = x - 4$** - Write as $y = x - 4$ - Swap $x$ and $y$: $x = y - 4$ - Solve for $y$: $y = x + 4$ - So, $f^{-1}(x) = x + 4$ **b) $f(x) = 6 - 5x$** - Write as $y = 6 - 5x$ - Swap $x$ and $y$: $x = 6 - 5y$ - Solve for $y$: $$x - 6 = -5y$$ $$\Rightarrow y = \frac{6 - x}{5}$$ - So, $f^{-1}(x) = \frac{6 - x}{5}$ **c) $f(x) = \frac{1}{2}x - 1$** - Write as $y = \frac{1}{2}x - 1$ - Swap $x$ and $y$: $x = \frac{1}{2}y - 1$ - Solve for $y$: $$x + 1 = \frac{1}{2}y$$ $$\Rightarrow y = 2(x + 1) = 2x + 2$$ - So, $f^{-1}(x) = 2x + 2$ **d) $f(x) = 3x + 1$** - Write as $y = 3x + 1$ - Swap $x$ and $y$: $x = 3y + 1$ - Solve for $y$: $$x - 1 = 3y$$ $$\Rightarrow y = \frac{x - 1}{3}$$ - So, $f^{-1}(x) = \frac{x - 1}{3}$ **e) $f(x) = \frac{3}{4}x + 2$** - Write as $y = \frac{3}{4}x + 2$ - Swap $x$ and $y$: $x = \frac{3}{4}y + 2$ - Solve for $y$: $$x - 2 = \frac{3}{4}y$$ $$\Rightarrow y = \frac{4}{3}(x - 2) = \frac{4}{3}x - \frac{8}{3}$$ - So, $f^{-1}(x) = \frac{4}{3}x - \frac{8}{3}$ **f) $f(x) = \frac{x - 3}{4}$** - Write as $y = \frac{x - 3}{4}$ - Swap $x$ and $y$: $x = \frac{y - 3}{4}$ - Solve for $y$: $$4x = y - 3$$ $$\Rightarrow y = 4x + 3$$ - So, $f^{-1}(x) = 4x + 3$