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!

The query submitted by the user: (fog)(x)=2x+10 and f(X)=x+2

Learn how to solve a function composition problem where (fog)(x) = 2x + 10 and f(x) = x + 2. This step-by-step guide explains how to find g(x) using key algebraic concepts like function composition, suitable for students in Grades 10-12.

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