Let's break down the problem from the image:

Given functions:

• $f(x) = x + 2$
• $g(x) = \sqrt{x - 3}$

We need to find the compositions:

1. $(f \circ g)(x) = f(g(x))$
2. $(g \circ f)(x) = g(f(x))$

1. Finding $(f \circ g)(x)$:

$f(g(x)) = f(\sqrt{x - 3})$

Substitute $\sqrt{x - 3}$ into $f(x)$: $f(\sqrt{x - 3}) = \sqrt{x - 3} + 2$

So, $(f \circ g)(x) = \sqrt{x - 3} + 2$

2. Finding $(g \circ f)(x)$:

$g(f(x)) = g(x + 2)$

Substitute $x + 2$ into $g(x)$: $g(x + 2) = \sqrt{(x + 2) - 3}$ $= \sqrt{x - 1}$

So, $(g \circ f)(x) = \sqrt{x - 1}$

Final Composed Functions:

1. $(f \circ g)(x) = \sqrt{x - 3} + 2$
2. $(g \circ f)(x) = \sqrt{x - 1}$

Do you want more details or have any questions about these steps?

Here are 5 related questions for further practice:

1. What is the domain of $(f \circ g)(x)$ and $(g \circ f)(x)$?
2. How do you find the composition $(h \circ f)(x)$ for another function $h(x)$?
3. If $f(x) = 2x$ and $g(x) = x^2 + 1$, what is $(f \circ g)(x)$?
4. How do you check if two functions are inverses using composition?
5. Can you compose three functions together, like $(f \circ g \circ h)(x)$?

Tip: When composing functions, always substitute the inner function into the outer function carefully, following the correct order of operations.