Question

Describe the vertical asymptote and holes for the graph of
Y=(x-3)(x-1)/(x-1)(x-5)

Answer 1

The function Y=(x-3)(x-1)/(x-1)(x-5), after simplification, reveals no holes, and the vertical asymptote is at x=5, which is where the function is undefined.

Step-by-step explanation:

The student is asking about the vertical asymptote and holes for the graph of the function Y=(x-3)(x-1)/(x-1)(x-5). First, we simplify the function to identify where the graph could potentially have holes or vertical asymptotes. By cancelling out the common factors in the numerator and the denominator, we get Y = (x-3)/(x-5). From this simplified expression, it is clear that there is no hole in the graph, since the (x-1) term that could have caused the hole is cancelled out.

Now, we examine the denominator to determine the vertical asymptote. Since the function is undefined when the denominator equals zero, setting (x-5) to zero gives us the x-coordinate of the vertical asymptote, which is x=5. Therefore, the vertical asymptote of this function is the line x=5.

Answer 2

Hole=1 and vertical asymptote=5

Step-by-step explanation:

It is given that:

`Y=((x-3)(x-1))/((x-1)(x-5))`

Look at the denominator and determine the values for which it is zero, therefore

(x-1)(x-5)=0⇒x=1,5

There is a common factor of (x-1), so we can cancel that common factor:

`Y=(x-3)/(x-5)`

But note we can only do that provided x-1≠0, x≠1.

We have already got that when x=1 the denominator is zero, so there is a removable discontinuity at that point (or a "hole").

when x=5, he denominator is also zero and this is the only vertical asymptote.