To find the range of the function $f(x) = \frac{2}{x^2 - 16}$, we will follow a systematic approach:

Step 1: Identify the domain

First, we need to consider the values of $x$ for which the denominator is non-zero. The denominator is $x^2 - 16$, so we set it equal to zero to find the points where the function is undefined:

$x^2 - 16 = 0$

$x^2 = 16$

$x = \pm 4$

Thus, $f(x)$ is undefined at $x = 4$ and $x = -4$, which means the domain of $f(x)$ is $x \in (-\infty, -4) \cup (-4, 4) \cup (4, \infty)$.

Step 2: Analyze the behavior of $f(x)$

Next, let's examine the function's behavior as $x$ approaches the values where it is undefined (i.e., $x = 4$ and $x = -4$):

• As $x \to 4$, the denominator $x^2 - 16$ approaches 0 from the positive side, making $f(x)$ approach $+\infty$.
• As $x \to -4$, the denominator $x^2 - 16$ also approaches 0 from the positive side, so $f(x)$ approaches $+\infty$.

As $|x| \to \infty$, the denominator $x^2 - 16$ grows large, causing $f(x)$ to approach 0 from the positive side. Therefore, $f(x) \to 0^+$ as $x \to \pm \infty$.

Step 3: Investigate the range

The function $f(x) = \frac{2}{x^2 - 16}$ is always positive since $x^2 - 16$ is always positive for $|x| > 4$. Thus, the range of $f(x)$ must be positive values.

• As $x \to 4^\pm$ or $x \to -4^\pm$, $f(x) \to \infty$.
• As $x \to \pm \infty$, $f(x) \to 0^+$, but never actually reaches 0.

Thus, the range of $f(x)$ is $(0, \infty)$.

Final Answer:

The range of $f(x) = \frac{2}{x^2 - 16}$ is $(0, \infty)$.

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Would you like further clarification on any of these steps or additional examples? Here are some related questions for you:

1. How do you find the range of a rational function?
2. What happens to the range of a function if it has a vertical asymptote?
3. How can you find the vertical asymptotes of rational functions like $f(x)$?
4. How does the behavior of a function change when you shift the graph horizontally or vertically?
5. How do you determine the horizontal asymptote of a rational function?

Tip: When finding the range of a rational function, look for points where the denominator can be zero (undefined points) and analyze the function’s behavior near these points.