Determine (fog)(x) y (g o f)(x)
1.f(x)=X elevado al cuadrado ; g(x)=1+x

To determine \((f \circ g)(x)\) and \((g \circ f)(x)\), where \(f(x) = x^2\) and \(g(x) = 1 + x\), let's proceed step by step.

### 1. \((f \circ g)(x)\)
This represents \(f(g(x))\). 

First, find \(g(x)\):
\[
g(x) = 1 + x
\]

Now, substitute \(g(x)\) into \(f(x)\):
\[
f(g(x)) = f(1 + x) = (1 + x)^2
\]

Expanding \((1 + x)^2\):
\[
(1 + x)^2 = 1 + 2x + x^2
\]

So, \((f \circ g)(x) = 1 + 2x + x^2\).

### 2. \((g \circ f)(x)\)
This represents \(g(f(x))\).

First, find \(f(x)\):
\[
f(x) = x^2
\]

Now, substitute \(f(x)\) into \(g(x)\):
\[
g(f(x)) = g(x^2) = 1 + x^2
\]

So, \((g \circ f)(x) = 1 + x^2\).

### Final Answer:
- \((f \circ g)(x) = 1 + 2x + x^2\)
- \((g \circ f)(x) = 1 + x^2\)

Would you like further details on any of these steps?

### Related Questions:
1. How do you determine \((f \circ g)(x)\) when \(f(x)\) is a trigonometric function?
2. What happens when both functions involve logarithms?
3. How do you approach composition when \(f(x)\) and \(g(x)\) are piecewise functions?
4. What are the conditions for \((f \circ g)(x)\) to be invertible?
5. How do you interpret the geometric meaning of composition of functions?

**Tip:** Remember that the order of composition matters: \((f \circ g)(x)\) is generally different from \((g \circ f)(x)\).

Learn how to solve function composition (f o g)(x) and (g o f)(x) using specific examples. Understand the relationship between quadratic functions and linear functions in function composition. Suitable for advanced high school students and those interested in advanced algebraic concepts.

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