AI Tutor
If you graph f(x)=x2βˆ’4xβˆ’2f(x) = \frac{x^2 - 4}{x - 2} on standard graphing software, it looks identical to the line y=x+2y = x + 2.
However, plugging in x=2x = 2 produces 00\frac{0}{0}, an undefined form that leaves a microscopic gap in the curve: an open point called a removable discontinuity (or hole).
How do you detect these invisible gaps algebraically without having to test every single number?

Canceling Factors to Find Holes

A rational function is a fraction formed by two polynomials, P(x)Q(x)\frac{P(x)}{Q(x)}. Any input that makes the denominator Q(x)=0Q(x) = 0 is excluded from the function's domain (its set of all valid inputs).
When a common binomial factor (xβˆ’c)(x - c) appears in both the numerator and the denominator, dividing it out simplifies the equation everywhere except at x=cx = c.
What happens when only some denominator factors cancel while others remain? Let's trace a full rational function to see the difference.

Worked Example: Locating Holes and Asymptotes

Find all discontinuities for the function f(x)=x2βˆ’xβˆ’6x2+xβˆ’2f(x) = \frac{x^2 - x - 6}{x^2 + x - 2}.
First, factor both quadratics completely: f(x)=(xβˆ’3)(x+2)(xβˆ’1)(x+2)f(x) = \frac{(x - 3)(x + 2)}{(x - 1)(x + 2)}
Setting the denominator equal to zero gives the domain restrictions xβ‰ 1x \neq 1 and xβ‰ βˆ’2x \neq -2.