For the function \( f(x)=\frac{x}{5}-9 \), find \( f^{-1}(x) \) Answer \( \begin{array}{l}f^{-1}(x)=5(x-9) \\ f^{-1}(x)=5(x+9) \\ f^{-1}(x)=\frac{(x+9)}{5} \\ f^{-1}(x)=\frac{x}{5}+9\end{array} \)

To find the inverse function \( f^{-1}(x) \) for \( f(x) = \frac{x}{5} - 9 \), start by replacing \( f(x) \) with \( y \), resulting in the equation: \[ y = \frac{x}{5} - 9. \] Next, solve for \( x \): 1. Add 9 to both sides: \( y + 9 = \frac{x}{5} \). 2. Multiply both sides by 5: \( 5(y + 9) = x \). Thus, the inverse function is: \[ f^{-1}(x) = 5(x + 9). \] Make sure to verify by substituting \( f^{-1}(x) \) back into \( f(x) \) and checking if you get \( x \) back! Now about the function's applications: In real life, inverse functions are crucial! For example, if \( f(x) \) represents the amount of money you have after spending based on your budget, \( f^{-1}(x) \) will help determine how much money you need to start with to afford certain expenditures. It's like knowing how much ice cream you can have after the bill!