If `f(x)=(x-2)/(x^(2)-4)` for what value(s) of x does the graph of f(x) have a vertical asymptote?

Options

A. `-2,0, and 2`
B. `-2 and 2`
C. `2`
D. `-2`

Explanation

To find the values of \( x \) for which the graph of the function \( f(x) = \frac{x - 2}{x^2 - 4} \) has a vertical asymptote, we need to follow these steps:

### Step 1: Identify when the denominator is zero
The vertical asymptotes of a function occur when the denominator is equal to zero and the numerator is not equal to zero.

The denominator of our function is:
\[ x^2 - 4 \]

### Step 2: Set the denominator equal to zero
We set the denominator equal to zero to find the points where the vertical asymptotes may occur:
\[ x^2 - 4 = 0 \]

### Step 3: Factor the denominator
The expression \( x^2 - 4 \) can be factored using the difference of squares:
\[ x^2 - 4 = (x - 2)(x + 2) \]

### Step 4: Solve for \( x \)
Now, we set each factor equal to zero:
1. \( x - 2 = 0 \) → \( x = 2 \)
2. \( x + 2 = 0 \) → \( x = -2 \)

### Step 5: Check the numerator
Next, we need to ensure that the numerator is not zero at these points. The numerator is:
\[ x - 2 \]

- For \( x = 2 \):
\[ x - 2 = 2 - 2 = 0 \quad \text{(not a vertical asymptote)} \]

- For \( x = -2 \):
\[ x - 2 = -2 - 2 = -4 \quad \text{(not zero)} \]

### Conclusion
Thus, the graph of \( f(x) \) has a vertical asymptote at:
\[ x = -2 \]

### Final Answer
The value of \( x \) for which the graph of \( f(x) \) has a vertical asymptote is:
\[ \boxed{-2} \]