(x^2+9)/(x-3)

Jasonbradley

3 min read

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Jun 17, 2024

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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.