Determine (fog)(x) y (g o f)(x)
f(x)=1/x+1
g(x)=raiz cuadrada de x + 1

Let's determine \((f \circ g)(x)\) and \((g \circ f)(x)\) given that:

- \(f(x) = \frac{1}{x} + 1\)
- \(g(x) = \sqrt{x} + 1\)

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

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

Now, substitute \(g(x)\) into \(f(x)\):
\[
f(g(x)) = \frac{1}{\sqrt{x} + 1} + 1
\]

So, \((f \circ g)(x) = \frac{1}{\sqrt{x} + 1} + 1\).

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

First, find \(f(x)\):
\[
f(x) = \frac{1}{x} + 1
\]

Now, substitute \(f(x)\) into \(g(x)\):
\[
g(f(x)) = \sqrt{\frac{1}{x} + 1} + 1
\]

So, \((g \circ f)(x) = \sqrt{\frac{1}{x} + 1} + 1\).

### Final Answer:
- \((f \circ g)(x) = \frac{1}{\sqrt{x} + 1} + 1\)
- \((g \circ f)(x) = \sqrt{\frac{1}{x} + 1} + 1\)

Would you like further details on any of these steps?

### Related Questions:
1. How do you simplify expressions involving square roots in compositions?
2. What is the domain of \((f \circ g)(x)\) and \((g \circ f)(x)\) in this case?
3. How would you approach the composition if \(f(x)\) and \(g(x)\) were polynomials?
4. Can you solve \((f \circ g)(x) = c\) for a given constant \(c\)?
5. What is the range of the composed functions \((f \circ g)(x)\) and \((g \circ f)(x)\)?

**Tip:** When working with compositions, carefully consider the domains of both functions to ensure the composed function is well-defined.

Explore the composition of functions f(x) = 1/x + 1 and g(x) = √x + 1. Learn how to compute (f ∘ g)(x) and (g ∘ f)(x) step-by-step, including domain considerations and range analysis. Suitable for advanced high school students and anyone interested in function theory.

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