Consider the function f (x) = 1 + 1/x . Express f (f (x)) as a single fraction, and simplify.

We are given the function:

\[
f(x) = 1 + \frac{1}{x}
\]

We want to express \( f(f(x)) \) as a single fraction and simplify it. Let's proceed step by step.

### Step 1: Substitute \( f(x) \) into itself.
We have \( f(x) = 1 + \frac{1}{x} \), so to find \( f(f(x)) \), we substitute \( f(x) \) into the function \( f \):

\[
f(f(x)) = 1 + \frac{1}{f(x)}
\]

### Step 2: Find \( \frac{1}{f(x)} \).
Since \( f(x) = 1 + \frac{1}{x} \), we need to find the reciprocal of \( f(x) \):

\[
\frac{1}{f(x)} = \frac{1}{1 + \frac{1}{x}}
\]

To simplify \( \frac{1}{1 + \frac{1}{x}} \), we need to get a common denominator in the denominator:

\[
\frac{1}{1 + \frac{1}{x}} = \frac{1}{\frac{x+1}{x}} = \frac{x}{x+1}
\]

### Step 3: Substitute \( \frac{1}{f(x)} \) back into \( f(f(x)) \).
Now that we have \( \frac{1}{f(x)} = \frac{x}{x+1} \), substitute this into the expression for \( f(f(x)) \):

\[
f(f(x)) = 1 + \frac{x}{x+1}
\]

### Step 4: Simplify the expression.
Now, we add the two terms:

\[
f(f(x)) = \frac{(x+1)}{(x+1)} + \frac{x}{x+1} = \frac{(x+1) + x}{x+1} = \frac{2x+1}{x+1}
\]

Thus, the simplified expression for \( f(f(x)) \) is:

\[
f(f(x)) = \frac{2x+1}{x+1}
\]

Would you like further explanation on any steps or additional details?

Here are some related questions to explore:
1. What happens if \( f(x) = \frac{1}{x} \)? How would \( f(f(x)) \) change?
2. Can you apply this method to a different function like \( f(x) = \frac{1}{x+1} \)?
3. How would you differentiate \( f(f(x)) \) using the chain rule?
4. How does the function \( f(f(x)) \) behave as \( x \to 0 \) or \( x \to \infty \)?
5. What are the fixed points of \( f(x) = 1 + \frac{1}{x} \)?

**Tip:** When dealing with complex fractions, always look to simplify by combining terms using a common denominator.

Consider the function f(x) = 1 + 1/x. Express f(f(x)) as a single fraction, and simplify.

This problem demonstrates how to express the composite function f(f(x)) where f(x) = 1 + 1/x as a single fraction. Detailed step-by-step simplification includes rational expressions and composite function operations. Ideal for students in Grades 10-12 studying algebra and functions.

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