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

4 min read Jun 17, 2024

Simplifying and Analyzing the Rational Expression (x-3)/(x^2-9)

The expression (x-3)/(x^2-9) is a rational expression, meaning it's a fraction where both the numerator and denominator are polynomials. Let's break down how to simplify it, identify its domain, and analyze its behavior.

Simplifying the Expression

The key to simplifying this expression is recognizing that the denominator can be factored. We can rewrite it as:

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

Now, we have a common factor of (x-3) in both the numerator and denominator. We can cancel this factor, resulting in:

1/(x+3)

This is the simplified form of the expression.

Domain of the Expression

The domain of a rational expression is all real numbers except for those that make the denominator zero. In this case, the denominator is zero when x = -3. Therefore, the domain of the expression is:

x ≠ -3

This means the expression is defined for all real numbers except for x = -3.

Analyzing the Expression

The simplified expression, 1/(x+3), tells us about the behavior of the original expression. Here are some key observations:

Vertical Asymptote: The expression has a vertical asymptote at x = -3. This means that as x approaches -3 from either side, the value of the expression approaches either positive or negative infinity.
Horizontal Asymptote: The expression has a horizontal asymptote at y = 0. This means that as x approaches positive or negative infinity, the value of the expression approaches zero.
Discontinuity: The expression has a discontinuity at x = -3. This is a result of the simplification process where we cancelled the (x-3) factor. This discontinuity is known as a hole because the expression is undefined at x = -3, but it approaches a finite value as x approaches -3.

Conclusion

By simplifying and analyzing the expression (x-3)/(x^2-9), we've gained a deeper understanding of its behavior. We've identified its simplified form, domain, vertical asymptote, horizontal asymptote, and discontinuity. These insights are crucial for graphing the expression, solving equations containing it, and understanding its role in various mathematical applications.