What is the inverse of \( f(x)=2 x^{3}-7 ? \) (1 point) \( f^{-1}(x)=\sqrt[3]{2 x-7} \) \( f^{-1}(x)=\sqrt[3]{2 x+7} \) \( f^{-1}(x)=\sqrt[3]{\frac{x-7}{2}} \) \( f^{-1}(x)=\sqrt[3]{\frac{x+7}{2}} \)

To find the inverse of the function \( f(x) = 2x^3 - 7 \), we start by replacing \( f(x) \) with \( y \): 1. Set \( y = 2x^3 - 7 \). 2. Solve for \( x \) in terms of \( y \): \[ y + 7 = 2x^3 \] \[ \frac{y + 7}{2} = x^3 \] \[ x = \sqrt[3]{\frac{y + 7}{2}} \] 3. Thus, the inverse function is \( f^{-1}(y) = \sqrt[3]{\frac{y + 7}{2}} \). So, we can write \( f^{-1}(x) = \sqrt[3]{\frac{x + 7}{2}} \). Here is the correct inverse from the options provided: \( f^{-1}(x) = \sqrt[3]{\frac{x + 7}{2}} \). Now, let’s sprinkle in some fun facts! Did you know that cubic functions like this one really pack a punch in calculus? They have interesting properties, such as inflection points, where the curvature changes! For the function \( 2x^3 - 7 \), there’s actually one inflection point located at \( x = 0 \), where the graph shifts its curvature. If you're ever solving for inverse functions, a common mistake is forgetting to swap \( x \) and \( y \) or messing up the arithmetic when isolating \( x \). Always double-check to ensure you're working with the correct terms and operations! They can sneak up on you during calculations, leading you down the wrong path in problem-solving. Happy calculating!