Question

Which of the following functions has a hole at (1,4)? ANSWER CHOICES IN THE IMAGE BELOW! PLEASE PROVIDE WORK WITH YOUR ANSWER:)!!! THANK YOU!

Answer 1

Hello!

The answer is:

d)
`((x-1)(11x+1))/((x-1)(x+2))`

Why?

A hole is a point where rational functions lose its continuity, meaning that in that point, there is a discontinuity condition.

We can find the hole of a rational function if there are similar terms on the numerator and the denominator by finding:

First (x-component): The values of x that makes the function equal to 0 in both numerator and denominator.

Second (y-component): Re-evaluating the same term in the other factors of the function to know the y-component.

Finding the x component we have:

`f(1)=((1-1)(11*1+1))/((1-1)(1+2))=((0)(12))/((0)(3))=(0)/(0)`

So, the x-component is 1,

Then, re-evaluating the function:

`f(1)=((x-1)(11*1+1))/((x-1)(1+2))=((12))/((3))=(12)/(3)=4`

Therefore, the y-component is 4,

Hence,

The function has a hole at (1,4)

Have a nice day!