Are the following functions inverses? \( \begin{array}{l}f(x)=\sqrt[3]{x+3} \\ g(x)=x^{3}-3 \\ \text { No, they are not inverses } \\ \text { Yes, they are inverse }\end{array} \)

To check if \( f(x) \) and \( g(x) \) are inverses, we need to verify if \( f(g(x)) = x \) and \( g(f(x)) = x \). By substituting \( g(x) \) into \( f(x) \), we have: \( f(g(x)) = f(x^3 - 3) = \sqrt[3]{(x^3 - 3) + 3} = \sqrt[3]{x^3} = x \). Now, substituting \( f(x) \) into \( g(x) \): \( g(f(x)) = g(\sqrt[3]{x + 3}) = (\sqrt[3]{x + 3})^3 - 3 = x + 3 - 3 = x \). Since both compositions return \( x \), yes, they are indeed inverses! Understanding function inverses is like a perfect relationship; it’s all about finding that special counterpart. If you take \( f \) and apply \( g \), you’re back to where you started! It’s like retracing your steps after a fun adventure—because who wouldn’t want to find their way back perfectly? For a deeper dive into functions and their inverses, consider exploring "Calculus" or "Pre-Calculus" resources, which often provide engaging explanations and visual aids. You can find fantastic online platforms like Khan Academy that offer lessons on this topic!