f(x) = x + 
1
x
,    g(x) = 
x + 11
x + 2   f ∘ f

To find \( (f \circ f)(x) \), we need to compute the composition of the function \( f(x) \) with itself. This means substituting \( f(x) \) into itself:

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

### Step-by-Step Calculation of \( (f \circ f)(x) \):

1. **Substitute \( f(x) \) into itself:**

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

2. **Apply the definition of \( f(x) \):**

To compute \( f\left( x + \frac{1}{x} \right) \), replace every \( x \) in the definition of \( f(x) \) with \( x + \frac{1}{x} \):

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

3. **Simplify the expression:**

Now, let's break this down further:

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

To simplify the expression \( \frac{1}{x + \frac{1}{x}} \), we need to rationalize the denominator:

\[
\frac{1}{x + \frac{1}{x}} = \frac{1}{\frac{x^2 + 1}{x}} = \frac{x}{x^2 + 1}
\]

4. **Combine the terms:**

Now, substitute this back into our expression for \( f(f(x)) \):

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

Combine like terms:

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

### Final Result:

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

Would you like more details on the steps, or have any questions?

Here are some related questions to explore further:

1. How do you find \( (g \circ g)(x) \) for the function \( g(x) \)?
2. What are the properties of function compositions?
3. How does the domain of \( f(x) \) affect \( (f \circ f)(x) \)?
4. Can you simplify the expression \( \frac{1}{x + \frac{1}{x}} \) further?
5. What are the potential applications of function compositions in real-world scenarios?

**Tip:** Remember that compositions of functions may have restrictions on their domains based on the original functions.

Explore how to compute (f ∘ f)(x) for the function f(x) = x + 1/x. This detailed solution covers the steps to find the composition of the function with itself, including rationalizing the denominator and simplifying the expression. Suitable for advanced high school students and anyone interested in function composition.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation