https://file.aiquickdraw.com/mathbot/file/1727182826051y47f9xlg.jpg

To solve the problem \( f \circ g \), which is the composition of two functions \( f(x) = x^2 - 4x \) and \( g(x) = \frac{1}{x+1} \), where \( x \neq -1 \), we need to substitute \( g(x) \) into \( f(x) \).

### Step-by-step solution:

1. The composition \( f \circ g \) means we substitute \( g(x) \) into \( f(x) \).
   \[
   (f \circ g)(x) = f(g(x))
   \]
2. Since \( f(x) = x^2 - 4x \), we substitute \( g(x) = \frac{1}{x+1} \) into \( f(x) \):
   \[
   f(g(x)) = \left( \frac{1}{x+1} \right)^2 - 4 \left( \frac{1}{x+1} \right)
   \]
3. Simplify each term:
   - First, \( \left( \frac{1}{x+1} \right)^2 = \frac{1}{(x+1)^2} \)
   - Second, \( 4 \left( \frac{1}{x+1} \right) = \frac{4}{x+1} \)
   
   So the expression becomes:
   \[
   f(g(x)) = \frac{1}{(x+1)^2} - \frac{4}{x+1}
   \]
4. To simplify further, find a common denominator:
   - The common denominator is \( (x+1)^2 \), so we rewrite the second term:
   \[
   \frac{4}{x+1} = \frac{4(x+1)}{(x+1)^2} = \frac{4x + 4}{(x+1)^2}
   \]
5. Now, subtract the two fractions:
   \[
   f(g(x)) = \frac{1 - (4x + 4)}{(x+1)^2} = \frac{1 - 4x - 4}{(x+1)^2} = \frac{-4x - 3}{(x+1)^2}
   \]
   
Thus, the composition \( (f \circ g)(x) \) is:
\[
(f \circ g)(x) = \frac{-4x - 3}{(x+1)^2}
\]

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

---

Here are 5 related questions:
1. How do we determine the domain of the composite function \( f \circ g \)?
2. Can we reverse the composition and find \( g \circ f \)?
3. What is the significance of finding the composition of two functions?
4. How can we graph the composite function \( (f \circ g)(x) \)?
5. What happens if \( g(x) \) is undefined, and how does that affect the composition?

**Tip:** When composing functions, always check the domain of each function involved to avoid undefined expressions.

Let f(x) = x^2 - 4x and g(x) = 1/(x + 1), x ≠ -1. Find f ∘ g.

Learn how to find the composition of functions where f(x) = x^2 - 4x and g(x) = 1/(x+1), including a detailed step-by-step solution using algebraic methods. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation