What is a characteristic event that defines the occurrence of holes in a function?

The occurrence of holes in a function is indicated by a factor that is present in both the numerator and the denominator of a rational function that can be simplified away. When such a factor equals zero, it leads to an undefined point in the function’s graph; however, if the factor can be canceled out, the function can still be evaluated at other points.

For instance, consider the rational function ( f(x) = \frac{(x-2)(x-3)}{(x-2)(x+1)} ). The factor ( (x-2) ) exists in both the numerator and the denominator. When you set ( x-2=0 ), you find that ( x=2 ) is a value where the function is undefined due to division by zero. However, due to the cancellation during simplification, ( f(x) ) behaves like a different function for all values other than ( x=2 ).

This indicates that ( x=2 ) creates a hole in the graph of the function, which is characterized by the shared factor. Thus, having a factor in both the numerator and denominator that equals zero at the same time is what definitively indicates the presence of holes in a