To find the range of the function $f(x) = \frac{x + 2}{x - 2}$, we need to determine the set of all possible output values (i.e., $f(x)$ values) for the inputs $x$ that are within the domain of the function.

Step 1: Identify the domain

The function $f(x) = \frac{x + 2}{x - 2}$ is defined for all $x$ except where the denominator is zero. The denominator is zero when: $x - 2 = 0 \implies x = 2$ So, the domain of $f(x)$ is all real numbers except $x = 2$.

Step 2: Find the range

To find the range, we set the function equal to a variable $y$ and solve for $x$: $y = \frac{x + 2}{x - 2}$

Multiplying both sides by $x - 2$: $y(x - 2) = x + 2$

Expanding and rearranging the equation: $yx - 2y = x + 2$ $yx - x = 2 + 2y$ $x(y - 1) = 2 + 2y$ $x = \frac{2 + 2y}{y - 1}$

The expression is valid for all real $y$ except when the denominator is zero, which occurs when $y - 1 = 0$. Thus: $y = 1$

Therefore, $y = 1$ is the only value that $f(x)$ cannot take. Thus, the range of the function is all real numbers except 1.

Final Answer

The range of $f(x) = \frac{x + 2}{x - 2}$ is: $\text{Range} = \mathbb{R} \setminus \{1\}$

Would you like more details or have any questions?

Here are some related questions:

1. What is the domain of the function $f(x) = \frac{x + 2}{x - 2}$?
2. How do you find the inverse of the function $f(x) = \frac{x + 2}{x - 2}$?
3. What are the asymptotes of $f(x) = \frac{x + 2}{x - 2}$?
4. How does $f(x) = \frac{x + 2}{x - 2}$ behave as $x \to \infty$?
5. How do you graph $f(x) = \frac{x + 2}{x - 2}$ using its critical points?

Tip: When finding the range, setting the function equal to $y$ and solving for $x$ helps identify values that $y$ cannot take.