What is the asymptotes for this rational function?

Question

determine if this funcition is a rational function if so find the domain and asymtotes. if the function is not rational, why?

f(x)= (2^x + 7) / (x^2 + x - 6)

Answer 1

it is a rational function because it's in the form of a fraction. you have to factor the denominator first, it's a trinomial and that is the only way to find out the asymtopes, X^2 + X-6 factors as     (x +3) (x-2), set it equal to 0

(x +3)=0                                   (x-2)=0

x+3=0                                        x-2=0

  -3   -3                                        +2   +2

x= -3                                          x=2

the domain is from:    

(negative infinity, -3) (-3, 2)  (2, positive infinity)

so the asymtopes are -3 and 2, remember to always factor the denominator if possible, how else could you possible know if you didn't factor? there's no way, you must factor. I  hope I was helpful :):)

Answer 2

there are 3 types of asymptotes...which are Vertical, Horizontal, and Oblique Linear asymptote.

to find the Vertical Asymptote, you need to make the denominator not = to 0, so tat means x^2 + x - 6 cannot = 0

therefore, the V.A is x= -3 and 2

to find the Horizontal Asymptote, just remember this rule :

If the degree of the denominator is higher than the degree of the numerator, then the H.A is just y = 0 which is the x-axis.

So in your case, the H.A, is just the x-axis.

Now for Oblique linear asymptote, this is the hard one. In this function, I don't think there is actually an O.L.A.

hope that helps =)