Describe the vertical asymptote(s) and hole(s) for the graph of .y=(x-3)(x-1)/(x-1)(x-5)

A. asymptote: x = 5 and hole: x = 1

B. asymptote: x = –5 and hole: x = –1

C. asymptote: x = –3 and hole: x = 5

D. asymptote: x = 5 and hole: x = –1

It’s A

The correct answer is A. The function has a vertical asymptote at x = 5, where the denominator becomes zero. It also has a hole at x = 1, where the numerator and denominator both become zero.

To find the vertical asymptotes and holes for the given function, we need to analyze the factors in the numerator and denominator separately.

In the numerator, we have (x - 3)(x - 1). The factors are (x - 3) and (x - 1).

In the denominator, we have (x - 1)(x - 5). The factors are (x - 1) and (x - 5).

To find the vertical asymptotes, we look for any values of x that make the denominator equal to zero (since division by zero is undefined). In this case, the denominator is equal to zero when (x - 1) = 0 or (x - 5) = 0. Solving these equations, we find that x = 1 and x = 5.

Therefore, the vertical asymptotes for the function are x = 1 and x = 5.

To determine the presence of any holes, we look for factors that are common to both the numerator and the denominator. In this case, we have the factor (x - 1) in both the numerator and the denominator.

When a factor is common to both the numerator and denominator, it creates a hole at the x-coordinate where the factor is equal to zero. In this case, the factor (x - 1) creates a hole at x = 1.

Therefore, the function has a vertical asymptote at x = 5 and a hole at x = 1.

Therefore, the correct answer is (A) asymptote: x = 5 and hole: x = 1.

is not b