1. **State the problem:** Solve for $x$ in the equation $y = f(x)$ where $f(x) = -5x + 1$. 2. **Write the equation:** $$y = -5x + 1$$ 3. **Isolate $x$:** Subtract 1 from both sides: $$y - 1 = -5x$$ 4. **Divide both sides by $-5$ to solve for $x$:** $$x = \frac{y - 1}{-5}$$ Show the cancellation step: $$x = \frac{\cancel{y - 1}}{\cancel{-5}}$$ 5. **Simplify the expression:** $$x = -\frac{y - 1}{5} = -\frac{y}{5} + \frac{1}{5}$$ --- 6. **Find the inverse of $f(x) = 7x + 12$:** Start with: $$y = 7x + 12$$ 7. **Swap $x$ and $y$ to find the inverse:** $$x = 7y + 12$$ 8. **Isolate $y$:** Subtract 12 from both sides: $$x - 12 = 7y$$ 9. **Divide both sides by 7:** $$y = \frac{x - 12}{7}$$ Show cancellation: $$y = \frac{\cancel{x - 12}}{\cancel{7}}$$ 10. **Write the inverse function:** $$g(x) = \frac{x - 12}{7}$$