https://file.aiquickdraw.com/mathbot/file/1725472264959qo7xzxdx.jpg

The problem involves finding the composition of two functions \( f(x) = \frac{3}{x+7} \) and \( g(x) = x^2 + 2 \). Elijah was asked to find \( f(g(x)) \), but his solution is incorrect.

### A. Identifying the Mistake
Let's examine the steps Elijah took:

- **Line 1:** \( f(g(x)) = f(x^2 + 2) \)
  - This line is correct. To find \( f(g(x)) \), you need to substitute \( g(x) = x^2 + 2 \) into the function \( f(x) \).
  
- **Line 2:** (missing in the image, but expected to be the correct substitution)
  - The correct next step should involve substituting \( x^2 + 2 \) into the function \( f(x) \). This would result in:
    \[
    f(g(x)) = f(x^2 + 2) = \frac{3}{(x^2 + 2) + 7} = \frac{3}{x^2 + 9}
    \]
  - But Elijah seems to have made a mistake.

- **Line 3:** \( \frac{3}{\frac{3}{x+7} + 3} \)
  - This is incorrect. Elijah seems to have incorrectly substituted back into the original function, creating an extra fraction, which was not required.

### **Where did Elijah make the mistake?**
Elijah made the mistake in **Line 3**. He incorrectly simplified or manipulated the expression after substituting \( g(x) \) into \( f(x) \).

### **Correct Approach:**
1. Start with \( f(g(x)) = f(x^2 + 2) \).
2. Substitute \( x^2 + 2 \) into the function \( f(x) \):
   \[
   f(g(x)) = \frac{3}{(x^2 + 2) + 7} = \frac{3}{x^2 + 9}
   \]

So, the correct solution should be:
\[
f(g(x)) = \frac{3}{x^2 + 9}
\]

### B. Show the Correct Way to Solve the Problem
Here is the correct step-by-step solution:

- **Step 1:** Identify \( g(x) = x^2 + 2 \).
- **Step 2:** Substitute \( g(x) \) into \( f(x) \):
  \[
  f(g(x)) = \frac{3}{x^2 + 2 + 7}
  \]
- **Step 3:** Simplify the expression:
  \[
  f(g(x)) = \frac{3}{x^2 + 9}
  \]

This is the correct final answer for \( f(g(x)) \).

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Do you have any questions, or would you like further details on this explanation?

### Related Questions:
1. What is the general method for finding the composition of two functions?
2. How do you correctly substitute one function into another?
3. Why is it important to simplify the expression correctly after substitution?
4. Can you explain a real-world scenario where function composition might be useful?
5. How does function composition relate to inverse functions?

**Tip:** Always double-check each step in function composition to ensure accuracy, especially when substituting complex expressions.

Given f(x) = 3/(x+7) and g(x) = x^2 + 2, Elijah was asked to find f(g(x)). He shared the following work:

Line 1: f(g(x)) = f(x^2 + 2)
Line 2: 
Line 3: 3 / ((3 / (x + 7)) + 3).

Learn how to correctly solve the function composition problem f(g(x)) where f(x) = 3/(x+7) and g(x) = x^2 + 2. This example highlights a common mistake made when substituting functions, and provides a step-by-step correction. Ideal for students in Grades 9-12.

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