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Given the function \( f(x) = \frac{1}{x^2 4} \), determine the vertical asymptotes of the function、To do this, follow these steps:
1、Identify the values of \( x \) that make the denominator zero.
2、Verify that these values also make the numerator non-zero.
To find the vertical asymptotes of the function \( f(x) = \frac{1}{x^2 4} \), we need to determine the values of \( x \) that make the denominator zero, as these values will cause the function to approach infinity or negative infinity.
Step 1: Identify the values of \( x \) that make the denominator zero.
The denominator of the function is \( x^2 4 \)、We set the denominator equal to zero and solve for \( x \):
\[ x^2 4 = 0 \]
This is a difference of squares, which can be factored as:
\[ (x 2)(x + 2) = 0 \]
Setting each factor equal to zero gives us:
\[ x 2 = 0 \quad \text{or} \quad x + 2 = 0 \]
Solving these equations, we get:
\[ x = 2 \quad \text{or} \quad x = -2 \]
Step 2: Verify that these values also make the numerator non-zero.
The numerator of the function is \( 1 \), which is a constant and does not depend on \( x \)、Therefore, the numerator is never zero for any value of \( x \).
Since the denominator is zero at \( x = 2 \) and \( x = -2 \) and the numerator is non-zero at these points, the function \( f(x) \) has vertical asymptotes at \( x = 2 \) and \( x = -2 \).
Thus, the vertical asymptotes of the function are:
\[ \boxed{x = 2 \text{ and } x = -2} \]