To analyze the function \( Y = \frac{x^2 - 4}{2 - x} \), let's break down the expression and explore its properties.

### Step 1: Simplify the numerator

First, notice that the numerator \( x^2 - 4 \) can be factored using the difference of squares:

\[ x^2 - 4 = (x - 2)(x + 2) \]

So, we can rewrite the function as:

\[ Y = \frac{(x - 2)(x + 2)}{2 - x} \]

### Step 2: Simplify the expression

Next, observe that \( 2 - x \) can be written as \( -(x - 2) \):

\[ 2 - x = -(x - 2) \]

Substitute this into the expression:

\[ Y = \frac{(x - 2)(x + 2)}{-(x - 2)} \]

Cancel the common factor \((x - 2)\) in the numerator and the denominator, remembering to account for the negative sign:

\[ Y = - (x + 2) \]

However, the cancellation of \( x - 2 \) implies that \( x \neq 2 \). So the function simplifies to:

\[ Y = -(x + 2) \quad \text{for} \quad x \neq 2 \]

### Step 3: Consider the domain

The function \( Y = \frac{x^2 - 4}{2 - x} \) is not defined when \( 2 - x = 0 \), which occurs when \( x = 2 \).

Thus, the simplified function \( Y = -(x + 2) \) is valid for all \( x \) except \( x = 2 \).

### Summary

The function \( Y = \frac{x^2 - 4}{2 - x} \) simplifies to:

\[ Y = -(x + 2) \]

with the restriction:

\[ x \neq 2 \]

Would you like further details or have any questions?

Here are 8 related questions to expand your understanding:

1. How do you factorize the difference of squares?
2. Why is it important to consider the domain when simplifying rational expressions?
3. Can you find the domain and range of the simplified function \( Y = -(x + 2) \)?
4. What happens to the graph of a function at points where the denominator is zero?
5. How do you handle simplifying expressions with common factors in the numerator and denominator?
6. Can you explain how to identify vertical asymptotes in rational functions?
7. How would the function change if the numerator were \( x^2 + 4 \) instead of \( x^2 - 4 \)?
8. Can you graph the function \( Y = \frac{x^2 - 4}{2 - x} \) and identify any discontinuities?

**Tip:** Always consider the domain of the original function before simplifying, especially identifying values that make the denominator zero to avoid undefined expressions.

Learn how to simplify the rational expression (x^2 - 4)/(2 - x) step-by-step. Understand factoring, domain restrictions, and the importance of considering the domain in rational expressions. Suitable for Grades 10-12.

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