We are given the function:

$f(x) = \frac{x^2 - 9}{x + 3}$

Step 1: Simplify the Function

First, we notice that the numerator $x^2 - 9$ is a difference of squares:

$x^2 - 9 = (x - 3)(x + 3)$

Thus, the function can be rewritten as:

$f(x) = \frac{(x - 3)(x + 3)}{x + 3}$

Step 2: Analyze Discontinuities

In this form, we can cancel out the factor $x + 3$ in the numerator and the denominator, but only when $x \neq -3$. This gives us the simplified form:

$f(x) = x - 3 \quad \text{for} \quad x \neq -3$

However, the original function has a denominator of $x + 3$, so $f(x)$ is undefined at $x = -3$ because division by zero occurs. Therefore, there is a discontinuity at $x = -3$.

Step 3: Type of Discontinuity

Since we can cancel out the common factor, the discontinuity at $x = -3$ is a removable discontinuity (or a hole in the graph).

Conclusion:

The function $f(x)$ has a removable discontinuity at $x = -3$.

Would you like a deeper explanation or have any further questions? Here are some follow-up questions:

1. How can we check if a discontinuity is removable or non-removable?
2. What is the graphical representation of a removable discontinuity?
3. How does factoring help in identifying discontinuities in rational functions?
4. What other types of discontinuities exist besides removable ones?
5. How do we evaluate limits near points of discontinuity?

Tip: Always check if you can factor the numerator and denominator to identify potential discontinuities in rational functions.