Question

Given the rational function f(x) = (3x^2 -7x+2)/(x^2 +4x +3) find all asympototes and intercepts

Answer 1

What is the limit of f(x) when x goes to infinity?

Answer 2

\[
\lim_{x\to \infty} (3x^2 -7x+2)/(x^2 +4x +3)=3
\]
Hence y=3 is horizontal asymptote.

Answer 3

The denominator can be factored
\[
x^2+4 x+3=(x+1) (x+3)
\]
Hence x=-1 and x=-3 are vertical Asymptotes.

Answer 4

The numerator can be factored as
\[
3 x^2-7 x+2=(x-2) (3 x-1)
\]
hence f(x) intercepts the x-axis on x=2 and x=1/3

Answer 5

If x=0, then y=2/3. that is the y-intercept.