1. **State the problem:** Find the inverse function $f^{-1}(x)$ of the function $f(x) = \frac{x}{7} + 5$ and expand any brackets in the answer. 2. **Recall the formula and rules:** To find the inverse function, we swap $x$ and $y$ in the equation and then solve for $y$. The original function is $y = \frac{x}{7} + 5$. 3. **Swap variables:** Replace $f(x)$ with $y$ and swap $x$ and $y$: $$x = \frac{y}{7} + 5$$ 4. **Solve for $y$:** Subtract 5 from both sides: $$x - 5 = \frac{y}{7}$$ Multiply both sides by 7: $$7(x - 5) = y$$ 5. **Expand brackets:** $$y = 7x - 35$$ 6. **Write the inverse function:** $$f^{-1}(x) = 7x - 35$$ **Explanation:** The inverse function reverses the effect of the original function. Since the original function divides by 7 and adds 5, the inverse multiplies by 7 and subtracts 35 to return to the original input. **Final answer:** $$f^{-1}(x) = 7x - 35$$