$$ f\left(x\right) $$=$$ \sqrt{x} $$, $$  $$g(x) = x+9, functions, find a. (f∘g)(x)(f∘g)(x) and b. (g∘f)(x)(g∘f)(x) Simplify

Let's solve the given problem step by step.

Given the functions:
- $ f(x) = \sqrt{x} $
- $ g(x) = x + 9 $

### Part a: $ (f \circ g)(x) $
This represents the composition of the functions $ f $ and $ g $, where $ f $ is applied after $ g $. Mathematically, it can be expressed as:

\[
(f \circ g)(x) = f(g(x))
\]

Substitute $ g(x) = x + 9 $ into $ f(x) = \sqrt{x} $:

\[
(f \circ g)(x) = f(x + 9) = \sqrt{x + 9}
\]

So, $ (f \circ g)(x) = \sqrt{x + 9} $.

### Part b: $ (g \circ f)(x) $
This represents the composition of the functions $ g $ and $ f $, where $ g $ is applied after $ f $. Mathematically, it can be expressed as:

\[
(g \circ f)(x) = g(f(x))
\]

Substitute $ f(x) = \sqrt{x} $ into $ g(x) = x + 9 $:

\[
(g \circ f)(x) = g(\sqrt{x}) = \sqrt{x} + 9
\]

So, $ (g \circ f)(x) = \sqrt{x} + 9 $.

### Final Answers:
a. $ (f \circ g)(x) = \sqrt{x + 9} $

b. $ (g \circ f)(x) = \sqrt{x} + 9 $

Would you like more details or have any questions?

Here are five related questions:
1. What is the domain of the function $ f(x) = \sqrt{x} $?
2. How do you find the domain of a composite function like $ (f \circ g)(x) $?
3. What happens to the range of $ g(x) = x + 9 $ when composed with $ f(x) = \sqrt{x} $?
4. How do you determine if a function is one-to-one or onto?
5. How can you graph the functions $ f(x) $ and $ g(x) $ and their compositions?

**Tip:** Always verify the domain and range of the composed function to ensure it makes sense within the context of the original functions.

Explore how to solve problems involving the composition of functions. Learn step-by-step solutions for (f ∘ g)(x) = √(x + 9) and (g ∘ f)(x) = √x + 9. Suitable for high school students studying functions and function composition.

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