The two functions you provided are:

1. $(f \circ g)(x) = 2x + 10$
2. $f(x) = x + 2$

To analyze this, I'll explain how to find $g(x)$ based on the composition of functions. The composition $(f \circ g)(x)$ means we apply $g(x)$ first and then apply $f(x)$ to the result. This can be written as:

$(f \circ g)(x) = f(g(x)) = 2x + 10$

Now, let's substitute $f(x) = x + 2$ into this equation:

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

So, we have:

$g(x) + 2 = 2x + 10$

To solve for $g(x)$, subtract 2 from both sides:

$g(x) = 2x + 8$

Thus, $g(x) = 2x + 8$.

Would you like further details on this? Here are some related questions:

1. How can we verify the result of a composition of functions?
2. What are the steps to solve for $g(x)$ from $(f \circ g)(x)$?
3. What happens when you change $f(x)$?
4. How does the concept of inverse functions apply to this problem?
5. What other types of function compositions are common in calculus?

Tip: When composing functions, always apply the innermost function first and then the outer function for accurate results!