Determine the Points of Discontinuity for the Function f(x) (3/x) * (x-1)/(x^2-1)

The function f(x) (3/x) * (x-1)/(x^2-1) is defined as a rational function, and its domain is all real numbers except where the denominator is zero. By identifying the points where the denominator becomes zero, we can determine the points of discontinuity.

Understanding Discontinuity and Asymptotes

A rational function is continuous on its entire domain if and only if it is defined at every point in that domain. For the given function, we need to find the values of x that make the denominator zero. These points are the discontinuities of the function.

The given function can be rewritten in a more structured form as:

$$ f(x) frac{3}{x} cdot frac{x-1}{x^2-1} $$

The denominator of this expression consists of two parts: frac{3}{x} and frac{x-1}{x^2-1}. We need to find the values of x that make both denominators zero.

Identifying Points of Discontinuity

For the first part, frac{3}{x}, the denominator is zero when x 0.

For the second part, frac{x-1}{x^2-1}, we solve the equation x^2 - 1 0 to find the points where the denominator is zero.

The equation x^2 - 1 0 can be factored as:

$$ x^2 - 1 (x - 1)(x 1) 0 $$

Solving for x, we find:

$$ x pm 1 $$

Thus, the function f(x) has discontinuities at x 0, x 1, and x -1.

Further Analysis of Discontinuities

Under the common terminology, there are asymptotes at the points where the function approaches infinity. Therefore, the points x 0 and x -1 are asymptotes for the function. On the other hand, the point x 1 is a removable singularity, meaning the function can be redefined at this point to make it continuous.

Verification Using a Common Denominator

To verify the points of discontinuity, we can rewrite the function using a common denominator:

$$ f(x) frac{3}{x} cdot frac{x-1}{x^2-1} frac{3x^2x-4}{x^3-x} $$

The new expression has a denominator (x^3 - x), which factors as:

$$ x^3 - x x(x^2 - 1) x(x-1)(x 1) $$

The roots of the polynomial (x^3 - x 0) are x 0, x 1, and x -1. These are the exact same points of discontinuity found earlier.

Conclusion

The function f(x) (3/x) * (x-1)/(x^2-1) has points of discontinuity at x 0, x 1, and x -1. The points x 0 and x -1 are asymptotes, while the point x 1 is a removable singularity. Understanding these points is crucial for analyzing the behavior of the function.