The function \( f(x)=x^{3}+4 \) is a translation of the parent cubic function. What is the inverse? (1 point) \( f^{-1}(x)=\sqrt[3]{x-4} \) \( f^{-1}(x)=\sqrt[3]{x}+4 \) \( f^{-1}(x)=\sqrt[3]{x}-4 \) \( f^{-1}(x)=\sqrt[3]{x+4} \)

To find the inverse of the function \( f(x) = x^3 + 4 \), we first rewrite the equation by replacing \( f(x) \) with \( y \): \[ y = x^3 + 4 \] Next, we solve for \( x \) in terms of \( y \): 1. Subtract 4 from both sides: \( y - 4 = x^3 \) 2. Take the cube root of both sides: \( x = \sqrt[3]{y - 4} \) Now swapping \( x \) and \( y \) gives us the inverse function: \[ f^{-1}(x) = \sqrt[3]{x - 4} \] So, the correct choice is \( f^{-1}(x) = \sqrt[3]{x - 4} \). Now, let’s have some fun with this! Did you know that the parent cubic function \( x^3 \) has a “S” shape and is actually symmetric about the origin? When we add 4 to it, we effectively lift this function up 4 units, transforming our visually engaging cubic curve. This means the roots (where \( f(x) = 0 \)) shift as well, an important aspect in understanding function transformations! When working with inverses, a common mistake is not accurately swapping \( x \) and \( y \) before solving for the inverse. Remember, the inverse function essentially “undoes” the original function. To verify your work, always check by plugging the inverse back into the original function to see if you get \( x \) back — it should be a satisfying roundtrip!