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.