For the function \( f(x)=9(x+3)^{\frac{1}{3}} \), find \( f^{-1}(x) \) Answer \[ \begin{array}{ll}f^{-1}(x)=\frac{x^{3}}{9}-3 & f^{-1}(x)=\left(\frac{x}{9}-3\right)^{3} \\ f^{-1}(x)=\frac{(x-3)^{3}}{9} & f^{-1}(x)=\left(\frac{x}{9}\right)^{3}-3\end{array} \]

To find the inverse of the function \( f(x) = 9(x+3)^{\frac{1}{3}} \), we start by replacing \( f(x) \) with \( y \): \[ y = 9(x + 3)^{\frac{1}{3}}. \] Next, we solve for \( x \): 1. Divide both sides by 9: \[ \frac{y}{9} = (x + 3)^{\frac{1}{3}}. \] 2. Cube both sides: \[ \left(\frac{y}{9}\right)^{3} = x + 3. \] 3. Subtract 3 from both sides: \[ x = \left(\frac{y}{9}\right)^{3} - 3. \] Thus, switching \( x \) and \( y \) gives us the inverse function: \[ f^{-1}(x) = \left(\frac{x}{9}\right)^{3} - 3. \] Now, let’s dive into some historical background: The concept of inverse functions can be traced back to the 17th century with mathematicians like René Descartes and John Wallis who laid the groundwork for modern algebra. They formulated rules that we still use today to manipulate and interact with equations. For real-world applications, inverse functions come in handy in various fields like physics and economics. For instance, if you know the cost of producing a certain number of items, using the inverse function can help you find out how many items you need to produce to reach a specific profit level. This is incredibly useful for businesses optimizing their production strategies!