f(x) = g(x) - 2s(x), g(x) = x ^ 2 + 2x, s(x) = e ^ x + 1

Given the following functions:

- \( f(x) = g(x) - 2s(x) \)
- \( g(x) = x^2 + 2x \)
- \( s(x) = e^x + 1 \)

We want to find the explicit expression for \( f(x) \).

### Step 1: Substitute the expressions for \( g(x) \) and \( s(x) \) into the formula for \( f(x) \):

\[
f(x) = (x^2 + 2x) - 2(e^x + 1)
\]

### Step 2: Simplify the expression:

\[
f(x) = x^2 + 2x - 2e^x - 2
\]

So, the final expression for \( f(x) \) is:

\[
f(x) = x^2 + 2x - 2e^x - 2
\]

This is the simplified form of the function \( f(x) \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What is the derivative of \( f(x) \)?
2. What is the integral of \( f(x) \)?
3. What are the roots of the equation \( f(x) = 0 \)?
4. What is the behavior of \( f(x) \) as \( x \to \infty \)?
5. What is the second derivative of \( f(x) \)?

**Tip:** Simplifying expressions early can help in solving complex calculus problems later on.

Learn how to solve the function composition problem where f(x) is defined as g(x) - 2s(x), with g(x) = x^2 + 2x and s(x) = e^x + 1. This problem explores function composition, exponential functions, and quadratic functions, providing a detailed step-by-step solution. Suitable for advanced high school students and those interested in calculus and function theory.

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