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

Given:
\[
g(x) = \frac{x + 11}{x + 2}
\]

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

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

\[
g(g(x)) = g\left( \frac{x + 11}{x + 2} \right)
\]

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

To compute \( g\left( \frac{x + 11}{x + 2} \right) \), we substitute \( \frac{x + 11}{x + 2} \) into the function \( g(x) \):

\[
g\left( \frac{x + 11}{x + 2} \right) = \frac{\left( \frac{x + 11}{x + 2} \right) + 11}{\left( \frac{x + 11}{x + 2} \right) + 2}
\]

3. **Simplify the expression:**

Now, simplify the numerator and the denominator separately.

- **Numerator:**

\[
\left( \frac{x + 11}{x + 2} \right) + 11 = \frac{x + 11 + 11(x + 2)}{x + 2} = \frac{x + 11 + 11x + 22}{x + 2} = \frac{12x + 33}{x + 2}
\]

- **Denominator:**

\[
\left( \frac{x + 11}{x + 2} \right) + 2 = \frac{x + 11 + 2(x + 2)}{x + 2} = \frac{x + 11 + 2x + 4}{x + 2} = \frac{3x + 15}{x + 2}
\]

4. **Combine the terms:**

Now, substitute these results back into our expression for \( g(g(x)) \):

\[
g(g(x)) = \frac{\frac{12x + 33}{x + 2}}{\frac{3x + 15}{x + 2}}
\]

Since both fractions have the same denominator \( x + 2 \), we can simplify by canceling out the denominators:

\[
g(g(x)) = \frac{12x + 33}{3x + 15}
\]

5. **Simplify the fraction:**

Factor out common terms from the numerator and the denominator:

\[
g(g(x)) = \frac{3(4x + 11)}{3(x + 5)} = \frac{4x + 11}{x + 5}
\]

### Final Result:

\[
(g \circ g)(x) = \frac{4x + 11}{x + 5}
\]

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 \( (f \circ g)(x) \) for the given functions \( f(x) \) and \( g(x) \)?
2. What are the conditions for the function \( g(x) \) to be invertible?
3. How does the domain of \( g(x) \) affect \( (g \circ g)(x) \)?
4. Can you find the fixed points of the function \( g(x) \)?
5. What happens if you compute \( (g \circ f)(x) \) instead?

**Tip:** When composing functions, it's crucial to check for restrictions in the domain caused by division by zero or other undefined operations.

Explore how to compute (g ∘ g)(x) for the function g(x) = (x + 11) / (x + 2). Step-by-step solution showing the process of function composition and simplification. Suitable for advanced high school and college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation