https://file.aiquickdraw.com/mathbot/file/17285247492213lb88jp4.jpg

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.

Find each of the following, if possible (if not possible, enter NONE): (a) f ∘ g (b) g ∘ f, where f(x) = √(x - 1) and g(x) = x^3 + 1.

Learn how to solve function compositions for f(x) = √(x - 1) and g(x) = x^3 + 1. The solution explores finding f ∘ g(x) and g ∘ f(x) and their domains. Suitable for high school students studying algebra.

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