Given the function $f(x) = \frac{5}{9 - x^2}$, we are tasked with finding its domain and range.

1. Domain:

The domain of a function consists of all real values of $x$ for which the function is defined. For rational functions (functions that have a denominator), the function is undefined wherever the denominator is zero.

Here, the denominator is $9 - x^2$. So, we set: $9 - x^2 \neq 0$ This implies: $x^2 \neq 9$ Taking square roots on both sides: $x \neq 3 \quad \text{and} \quad x \neq -3$ Thus, the function is undefined at $x = 3$ and $x = -3$.

So, the domain is: $\boxed{(-\infty, -3) \cup (-3, 3) \cup (3, \infty)}$

2. Range:

The range of a function consists of all possible output values (or $y$-values) of the function.

To find the range, we consider the form of the function: $f(x) = \frac{5}{9 - x^2}$

Since the denominator $9 - x^2$ cannot be zero, $9 - x^2$ must be either positive or negative but never zero. The denominator decreases as $x^2$ increases, and the function's behavior changes accordingly.

Key points:

• As $x^2 \to 9$ (as $x$ approaches 3 or -3), the denominator $9 - x^2 \to 0$, making the function tend toward infinity.
• As $x^2$ becomes very large (far from 3 and -3), the denominator becomes more negative, and the overall function tends towards zero but is always negative for large $x^2$.

Thus, the range of $f(x)$ is: $\boxed{(-\infty, 0) \cup (0, \infty)}$ (Note that $0$ is excluded because the function never actually reaches $0$.)

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Would you like a detailed breakdown of how the range is derived or have any other questions?

Related questions:

1. How do you find the domain of a square root function?
2. What is the domain of $f(x) = \frac{1}{x - 2}$?
3. How do asymptotes affect the range of rational functions?
4. Can a function have a domain that excludes more than two points?
5. How would the range change if the numerator in this function were different?

Tip: When determining the domain of rational functions, always check where the denominator is zero—these are points of discontinuity.