Let f : R → R. Find the domain and range of the following functions:
(i) 5
9−x2 

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 \( 0 \).)

---

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.

Let f : R → R. Find the domain and range of the following functions: (i) 5 / (9 − x^2)

Understand how to find the domain and range of the rational function f(x) = 5 / (9 − x^2). This problem explores key concepts in algebra, such as identifying discontinuities and asymptotes. Suitable for students in Grades 10-12.

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