## Understanding the Expression (x^2 + 9) / (x - 3)

The expression (x^2 + 9) / (x - 3) represents a **rational function**. This means it's a function where the numerator and denominator are both polynomials. Let's break down its components and explore its key properties.

### The Numerator: x^2 + 9

**Quadratic Expression:**This is a quadratic expression, meaning it has a highest power of 2.**Irreducible:**It can't be factored further into real linear factors. This is because the expression has no real roots.

### The Denominator: x - 3

**Linear Expression:**This is a linear expression, meaning it has a highest power of 1.**Zero at x = 3:**The denominator becomes zero when x = 3. This is important because it signifies a**vertical asymptote**in the graph of the function.

### Important Points to Note:

**Domain:**The domain of this function is all real numbers**except**for x = 3. This is because division by zero is undefined.**Vertical Asymptote:**The vertical line x = 3 represents a vertical asymptote. This means the function approaches infinity as x approaches 3 from either side.**No Horizontal Asymptote:**Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote.**Behavior as x Approaches Infinity:**As x approaches positive or negative infinity, the function approaches positive infinity. This is because the numerator grows faster than the denominator.

### Analyzing the Graph

The graph of (x^2 + 9) / (x - 3) will have the following characteristics:

- A vertical asymptote at x = 3.
- The function will approach infinity as x approaches 3 from both sides.
- The function will increase without bound as x approaches positive or negative infinity.

This means the graph will have a shape similar to a hyperbola, with the vertical asymptote acting as a dividing line.

### Conclusion

(x^2 + 9) / (x - 3) represents a rational function with a vertical asymptote at x = 3. Understanding its components and key properties allows us to analyze its behavior and visualize its graph effectively.