The given problem is:

• $f(x) = x + 7$
• $(f \circ g)(x) = 5x^2 + 3$
• We need to determine $g(2)$.

Solution:

1. The notation $(f \circ g)(x)$ represents the composition of functions, meaning $(f \circ g)(x) = f(g(x))$.

2. Using the given information $(f \circ g)(x) = 5x^2 + 3$, we can substitute this into the composition form: $f(g(x)) = 5x^2 + 3$

3. The function $f(x) = x + 7$, so applying it to $g(x)$ gives: $f(g(x)) = g(x) + 7$ Thus, $g(x) + 7 = 5x^2 + 3$

4. Solve for $g(x)$: $g(x) = 5x^2 + 3 - 7 = 5x^2 - 4$

5. To find $g(2)$, substitute $x = 2$ into the expression for $g(x)$: $g(2) = 5(2)^2 - 4 = 5(4) - 4 = 20 - 4 = 16$

Final Answer:

$g(2) = 16$.

Let me know if you need further clarification! Here are some related questions:

1. What does function composition mean in general?
2. Can we use this same approach if $f(x)$ were a more complex function?
3. How would the solution change if we were asked for $g(3)$ instead?
4. What are the properties of polynomial functions like $g(x) = 5x^2 - 4$?
5. How can we verify the correctness of our composition in this context?

Tip: When dealing with function composition, always focus on understanding how the outer function applies to the result of the inner function.