What is the Point of Discontinuity of the Function f(x) (x^2 - 4) / (x - 2x - 3)?

Understanding the points of discontinuity in a function is crucial for comprehending its behavior. This article delves into the analysis of the function (f(x) frac{x^2 - 4}{x - 2x - 3}). We will simplify the function, determine the points of discontinuity, and explore the nature of these discontinuities.

Simplification and Basic Analysis

First, let's simplify the given function by combining like terms in the denominator.

Step 1: Combining Like Terms

Given: (f(x) frac{x^2 - 4}{x - 2x - 3})

Combine like terms in the denominator:

(f(x) frac{x^2 - 4}{-x - 3})

Further simplify to:

(f(x) frac{x^2 - 4}{x - 3})

Identifying Points of Discontinuity

A rational function is discontinuous whenever the denominator is equal to zero because division by zero is undefined. To find the points of discontinuity, set the denominator equal to zero and solve for x.

Set the denominator (x - 3 0) and solve for x:

(x 3)

Therefore, the function (f(x)) is discontinuous at (x 3).

Nature of the Discontinuity

To determine the nature of the discontinuity at (x 3), we examine the factors of the numerator and denominator.

Given: (f(x) frac{(x 2)(x - 2)}{x - 3})

The factors of the numerator are (x 2) and (x - 2), while the factor of the denominator is (x - 3). Since the denominator is (x - 3), and it does not share any common factors with the numerator, the discontinuity at (x 3) is essential.

Removable Discontinuity at (x 2)

However, there is also another point of discontinuity at (x 2). This discontinuity is removable.

Consider the original function before simplification:

(f(x) frac{(x 2)(x - 2)}{(x - 2)(x 1)})

This simplifies to:

(f(x) frac{x 2}{x 1}) for all (x eq 2)

Therefore, we can redefine (f(x)) to be (frac{x 2}{x 1}) at (x 2) to remove the discontinuity.

Asymptotes

It is also worth noting that the given function has asymptotes at (x 3) and (y 1). The vertical asymptote occurs at (x 3) because the denominator approaches zero there, and the horizontal asymptote at (y 1) is determined by analyzing the behavior of the function as (x) approaches infinity.

Conclusion

The function (f(x) frac{x^2 - 4}{x - 2x - 3}) has two points of discontinuity: an essential discontinuity at (x 3) and a removable discontinuity at (x 2). Understanding these points of discontinuity is crucial for a comprehensive analysis of the function's behavior.