To find the composite function \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \).

Given:
- \( g(x) = x - 5 \)
- \( f(x) = x^2 + 3 \)

Now, for the composite function \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(x - 5)
\]

Since \( f(x) = x^2 + 3 \), replace \( x \) in \( f(x) \) with \( x - 5 \):
\[
f(g(x)) = (x - 5)^2 + 3
\]

Now, expand \( (x - 5)^2 \):
\[
(x - 5)^2 = x^2 - 10x + 25
\]

Thus:
\[
f(g(x)) = x^2 - 10x + 25 + 3 = x^2 - 10x + 28
\]

So, the composite function is:
\[
f(g(x)) = x^2 - 10x + 28
\]

Would you like more details or have any questions about this process?

Here are 5 related questions to expand this topic:
1. How do you find the composite function if \( f(x) \) and \( g(x) \) are different?
2. What is the domain of the composite function \( f(g(x)) \)?
3. How would the composite function change if \( f(x) = x^2 - 3 \)?
4. What is the difference between \( f(g(x)) \) and \( g(f(x)) \)?
5. How does the order of composition affect the resulting function?

**Tip**: Always check if the functions are defined for all \( x \) in the domain before composing them.

Learn how to find the composite function f(g(x)) when f(x) = x^2 + 3 and g(x) = x - 5. This solution includes step-by-step expansion of the quadratic expression and is suitable for students in Grades 9-12.

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