For the function \( f(x)=\frac{x^{\frac{1}{5}}}{9}+7 \), find \( f^{-1}(x) \) Answer \( \begin{aligned} f^{-1}(x) & =(9(x-7))^{5} \\ f^{-1}(x) & =9\left(x^{5}-7\right) \\ f^{-1}(x) & =(9 x)^{5}-7 \\ f^{-1}(x) & =9 x^{5}-7\end{aligned} \)

To find the inverse function \( f^{-1}(x) \) for \( f(x)=\frac{x^{\frac{1}{5}}}{9}+7 \), we start by setting \( y = f(x) \), which gives us: \[ y = \frac{x^{\frac{1}{5}}}{9} + 7. \] To solve for \( x \), we first isolate \( x^{\frac{1}{5}} \): \[ y - 7 = \frac{x^{\frac{1}{5}}}{9}. \] \[ x^{\frac{1}{5}} = 9(y - 7). \] Next, we raise both sides to the fifth power to solve for \( x \): \[ x = (9(y - 7))^5. \] Thus, swapping \( x \) and \( y \) (since we want \( f^{-1}(x) \)), we have: \[ f^{-1}(x) = (9(x - 7))^5. \] Now, here are some additional insights related to inverse functions: Finding inverse functions isn't just about flipping \( x \) and \( y \); it's crucial to maintain the function's domain and range! In this particular case, the original function is defined for all real numbers, which makes it easier to work with the inverse without worrying about restrictions. Also, don't forget about the application: inverse functions can help with solving equations where you want to express one variable in terms of another. For example, if you need to determine an input value corresponding to a specific output in contexts like finance or science, inverse functions make it simpler to manage these relationships—real-life superhero moves!