This is a mathematical explanation of a given function $f(x) = \frac{x^2 - 9}{x - 3}$. Let’s break down the steps:

1. Domain of $f(x)$:

   • The function $f(x)$ is defined as a rational function. It involves division, so $x \neq 3$ to avoid division by zero.
   • Thus, the domain of $f(x)$ is $\mathbb{R} - \{3\}$, meaning all real numbers except 3.

2. Simplification of $f(x)$:

   • $x^2 - 9$ is factored as $(x - 3)(x + 3)$, leading to: $f(x) = \frac{(x - 3)(x + 3)}{x - 3}.$
   • For $x \neq 3$, the $(x - 3)$ terms cancel out, leaving: $f(x) = x + 3.$

3. Range of $f(x)$:

   • After simplification, $f(x) = x + 3$ for $x \neq 3$.
   • Substituting $x = 3$ into $x + 3$ gives 6, but since $x = 3$ is excluded from the domain, $f(x) = 6$ is not achievable.
   • Therefore, the range of $f(x)$ is $\mathbb{R} - \{6\}$, which means all real numbers except 6.

Summary:

• Domain: $\mathbb{R} - \{3\}$
• Range: $\mathbb{R} - \{6\}$

Would you like me to provide a graph of this function or clarify further steps?

Here are five related questions for deeper understanding:

1. What happens to the function's range if $x^2 - 9$ is replaced with another polynomial?
2. How can you determine the range of other rational functions with similar structures?
3. What is the significance of a removable discontinuity in a function?
4. Can we graphically confirm the excluded value from the range?
5. How does the cancellation of terms in rational functions impact the overall function?

Tip: Always check for discontinuities in rational functions by analyzing the denominator for zero values.