The function you're working with is $f(x) = \frac{x^2 - 4}{x - 2}$, and you're simplifying it. Let's go through the steps carefully:

1. Factor the numerator:
   The expression $x^2 - 4$ is a difference of squares, which can be factored as: $x^2 - 4 = (x - 2)(x + 2)$ So the function becomes: $f(x) = \frac{(x - 2)(x + 2)}{x - 2}$

2. Simplify the expression:
   As long as $x \neq 2$, we can cancel the $(x - 2)$ terms from the numerator and denominator: $f(x) = x + 2$

   However, you must be careful about domain restrictions. The original expression $\frac{x^2 - 4}{x - 2}$ is undefined at $x = 2$, because the denominator would be zero at this point.

Thus, the simplified function is: $f(x) = x + 2, \quad \text{for} \quad x \neq 2$

To summarize:

• The simplified function is $f(x) = x + 2$.
• However, $f(x)$ is undefined at $x = 2$ due to the original denominator.

Would you like to go deeper into understanding the domain of the function or more details?

5 Related Questions:

1. What is the domain and range of the function $f(x) = \frac{x^2 - 4}{x - 2}$?
2. How do we determine if a function has a hole in its graph?
3. Can you explain how limits relate to the simplification of this function at $x = 2$?
4. What happens to the graph of $f(x) = \frac{x^2 - 4}{x - 2}$ near $x = 2$?
5. Why can't we cancel terms if $x = 2$?

Tip:

Always remember to check for restrictions on the domain when simplifying rational expressions to avoid canceling terms that lead to undefined points.