what is the range of this equation f(x)=x+2/x-2

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.

What is the range of this equation f(x) = (x + 2) / (x - 2)?

Explore how to find the range of the rational function f(x) = (x + 2) / (x - 2). This detailed solution guides you through the process of solving for the range, excluding the value y = 1. Ideal for high school students in Grades 9-12 studying algebra and functions.

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