Answers to writing a rational function with steps

Question

(a) Vertical asymptote: x=1

horizontal asymptote: y=0

zero: x=4

f(x)=?

b) Vertical asymptote: x=-2

horizontal asymptote: y=0

zero: x=1

f(x)=?

c) Vertical asymptote: x= -1 , x=2

horizontal asymptote: y=3

zero: x=4 , x= -4

f(x)=?

d) Vertical asymptote: x= -5 , x=2

horizontal asymptote: y= -3

zero: x= -2 , x= -4

f(x)=?

Answer 1

A) vertical asymptote occurs where the denominator is zero so (x-1) is in denominator

The function equals zero when the numerator equals zero so (x-4) is I’m the numerator

For the horizontal asymptote to equal zero the degree of the denominator must be > the degree of the numerator, so have to have another x in the denominator.

This gives us the function (x-4)/(x)(x-1)

B) using the same logic this is (x-1)/(x)(x+2)

C) If the horizontal asymptote is 3 then the degree of the numerator and denominator must be equal and their ratio is 3:1 so we have 3(x-4)(x+4)/(x+1)(x-2)

I will leave D for you to do