The problem in the image asks to find the following compositions of functions, given the definitions:

• $f(x) = \sqrt{x - 1}$
• $g(x) = x^3 + 1$

You are tasked with finding $f \circ g$ and $g \circ f$.

(a) $f \circ g(x)$

This means applying $g(x)$ first, then applying $f(x)$ to the result: $f \circ g(x) = f(g(x)) = f(x^3 + 1)$ Now substitute $x^3 + 1$ into $f(x)$: $f(x^3 + 1) = \sqrt{(x^3 + 1) - 1} = \sqrt{x^3}$ So, $f \circ g(x) = \sqrt{x^3}$ This simplifies to $|x^{3/2}|$, but in many cases, $\sqrt{x^3}$ will be an acceptable answer.

(b) $g \circ f(x)$

This means applying $f(x)$ first, then applying $g(x)$ to the result: $g \circ f(x) = g(f(x)) = g(\sqrt{x - 1})$ Now substitute $\sqrt{x - 1}$ into $g(x)$: $g(\sqrt{x - 1}) = (\sqrt{x - 1})^3 + 1 = (x - 1)^{3/2} + 1$ So, $g \circ f(x) = (x - 1)^{3/2} + 1$

These are the compositions of the two functions.

Would you like a detailed explanation or have any questions about these steps? Here are some related questions for further practice:

1. What is the domain of $f \circ g(x)$?
2. What is the domain of $g \circ f(x)$?
3. How would you compute $f(f(x))$?
4. How would you compute $g(g(x))$?
5. What would happen if $g(x) = \sqrt{x} + 1$? How would that affect the compositions?

Tip: When working with function composition, always check the domains of both functions to ensure that the composition is defined.