Consider the following rational functions: r(x)=(2x-1)/(x^2-x-2). s(x)=(x^3+27)/(x^2+4) t(x)=(x^3-9x)/(x+2) u(x)=((x^2)+x-6)/(x^2-25) w(x)=(x^3+6x^2+9x)/(x+3) What are the asymptotes of the function r(x)?

geduiwelh

geduiwelh

Answered question

2021-05-03

Consider the following rational functions: r(x)=2x1x2x2.
s(x)=x3+27x2+4
t(x)=x39xx+2
u(x)=(x2)+x6x225
w(x)=x3+6x2+9xx+3
What are the asymptotes of the function r(x)?

Answer & Explanation

Khribechy

Khribechy

Skilled2021-05-04Added 100 answers

Work as shown below, follow the steps:
r(x)=2x1x2x2
- For the vertical asymptotes find the zerosofthe polynomialin the denominator
x2x2=0 [write —x as x — 2x]
x2+x2x2=0 [group terms 1st-3rd and 2nd-4th]
-> x(x-2)+(x-2)=0 [factor out (x — 2)]
-> (x-2)(x+1)=0 [zero product property]
-> x=2, x=-1 The vertical asymptotes
- For the horizontal asymptote, compare the degrees of the polynomials in the numerator and the denominator:
The degree of the polynomial in the numerator 1 is less than the degree of the polynomial in the denominator 2 so the horizontal asymptote of the graph of this function is the line:
y=0 (the x-axis) The horizontal asymptote

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