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Let's go through the problem step by step.

### Given Functions:
- \( f(x) = \sqrt{x} \)
- \( g(x) = x + 9 \)

### (a) Find \( (f \circ g)(x) \):
This is the composition \( f(g(x)) \).

- \( g(x) = x + 9 \), so \( f(g(x)) = f(x + 9) = \sqrt{x + 9} \).
- The simplified result is \( \sqrt{x + 9} \).

**Domain of \( (f \circ g)(x) \):**

For \( f(g(x)) = \sqrt{x + 9} \), the expression under the square root must be non-negative:
- \( x + 9 \geq 0 \)
- \( x \geq -9 \)

Thus, the domain of \( f(g(x)) \) is \( [-9, \infty) \).

### (b) Find \( (g \circ f)(x) \):
This is the composition \( g(f(x)) \).

- \( f(x) = \sqrt{x} \), so \( g(f(x)) = g(\sqrt{x}) = \sqrt{x} + 9 \).
- The simplified result is \( \sqrt{x} + 9 \).

**Domain of \( (g \circ f)(x) \):**

For \( g(f(x)) = \sqrt{x} + 9 \), the square root \( \sqrt{x} \) requires \( x \geq 0 \).

Thus, the domain of \( g(f(x)) \) is \( [0, \infty) \).

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### Corrected Results:
- \( (f \circ g)(x) = \sqrt{x + 9} \), with domain \( [-9, \infty) \).
- \( (g \circ f)(x) = \sqrt{x} + 9 \), with domain \( [0, \infty) \).

Would you like further details on the steps, or do you have any questions? 

Here are 5 related questions for deeper understanding:
1. How do you determine the domain of a square root function?
2. What is the difference between \( f \circ g \) and \( g \circ f \)?
3. How would the domain change if the square root were replaced by a cubic root?
4. Can you compose functions when one is not defined at certain points?
5. How does the range of \( f(x) = \sqrt{x} \) affect the domain of \( g(f(x)) \)?

**Tip:** When composing functions, always ensure that the output of the inner function fits the domain of the outer function.

Consider the pair of functions f(x) = √x and g(x) = x + 9. Find (f ∘ g)(x) and (g ∘ f)(x). Simplify the results and determine the domain of each new function.

Learn how to compose functions with square roots and linear equations. This problem demonstrates how to find the composition of f(x) = √x and g(x) = x + 9, simplify the results, and determine the domains. Suitable for high school students studying function composition.

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