Questions: If f(x) = x^3 - 1, g(x) = x - 6, and h(x) = sqrt(x), find the following: a) f(g(h(x)))

Solution Steps

To solve the problem, we need to find the composition of the functions \( f \), \( g \), and \( h \). Specifically, we need to evaluate \( f(g(h(x))) \). This involves the following steps:

1. Compute \( h(x) \).
2. Substitute \( h(x) \) into \( g(x) \) to get \( g(h(x)) \).
3. Substitute \( g(h(x)) \) into \( f(x) \) to get \( f(g(h(x))) \).

Given \( h(x) = \sqrt{x} \), we start by evaluating \( h(x) \).

Next, we substitute \( h(x) \) into \( g(x) \). Given \( g(x) = x - 6 \), we have: \[ g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 6 \]

Finally, we substitute \( g(h(x)) \) into \( f(x) \). Given \( f(x) = x^3 - 1 \), we have: \[ f(g(h(x))) = f(\sqrt{x} - 6) = (\sqrt{x} - 6)^3 - 1 \]

Final Answer