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To solve the given problem, we need to find \( f(g(x)) \) or \( (f \circ g)(x) \), which means composing the functions \( f(x) \) and \( g(x) \).

We are provided with the following functions:
- \( f(x) = x^2 + 2x \)
- \( g(x) = x + 3 \)

The composition \( f(g(x)) \) involves substituting the function \( g(x) \) into the function \( f(x) \).

Let's compute \( f(g(x)) \):

1. Start with \( g(x) = x + 3 \).
2. Substitute \( g(x) \) into \( f(x) \), which means replacing every \( x \) in the function \( f(x) \) with \( g(x) = x + 3 \).

Thus, \( f(g(x)) = (x + 3)^2 + 2(x + 3) \).

Now, let's expand and simplify this expression.The composition \( f(g(x)) \) simplifies to:

\[
f(g(x)) = x^2 + 8x + 15
\]

Would you like any further explanation or details on this solution?

Here are five related questions to expand on this topic:
1. How would you find \( g(f(x)) \)?
2. What is the domain of the function \( f(g(x)) \)?
3. How can you find the inverse of the composed function \( f(g(x)) \)?
4. What are the key differences between function composition and multiplication of functions?
5. Can you graph both \( f(x) \) and \( f(g(x)) \) on the same coordinate plane?

**Tip:** When composing functions, always replace the variable in the outer function with the entire expression of the inner function.

Find f(g(x)) or (f ∘ g)(x) where f(x) = x^2 + 2x and g(x) = x + 3

Learn how to compose functions by finding f(g(x)) where f(x) = x^2 + 2x and g(x) = x + 3. This algebraic problem covers key concepts such as expanding and simplifying expressions.

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