The function \( h(x)=(x+8)^{6} \) can be expressed in the form \( f(g(x)) \), where \( f(x)=x^{6} \), and \( g(x) \) is defined below: \( g(x)=\square \)

To express \( h(x)=(x+8)^{6} \) in the form \( f(g(x)) \) with \( f(x)=x^{6} \), we need to identify \( g(x) \). Since \( f(g(x)) \) implies that \( g(x) \) is the input to the function \( f \), we can see that in this scenario, \( g(x) \) must be the expression contained within the parentheses of \( (x+8)^{6} \). Thus, we find that \( g(x) = x + 8 \). This means \( h(x) = f(g(x)) = f(x + 8) = (x + 8)^{6} \). To sum it up: In this formulation, \( g(x) \) serves as a transformation of the input \( x \) that modifies it before it's elevated to the sixth power by \( f \). The elegance of composing functions allows us to see how mathematical operations can be intertwined seamlessly! Beyond just breaking it down, this kind of function composition is found everywhere in calculus and data science, helping to model complex behaviors simply and effectively. It’s like combining different ingredients to create a delicious dish – simple components can create something extraordinary when combined!