Answer:
(10x ^ 2 - 70 * x +120) / (x ^ 2 + 7 * x +12)
Step-by-step explanation:
First a rational function is a function that is in the form a / b.
Knowing this, we proceed to calculate the asymptotes in x.
Vertical asymptotes
So that there is an asymptote in x, in a rational function the easiest thing is that the denominator is 0. Therefore, for the asymptotes x = -3 and x = -4, we make it 0 in both cases. That is to say:
x + 3 = 0
x + 4 = 0
both, in the denominator. For the moment it would be something like this:
1 / (x + 3) * (x + 4)
X Intersections
Now, for intersections, what we should do equation 0, and that the solution of x us of the value we want, in this case x = 3 and x = 4. Because here the denominator does not influence because when equal to 0 to a rational function, the denominator becomes 0. Similarly, it is equal to 0 as in the previous case, only this time they would go in the numerator. It would be:
x - 3 = 0
x - 4 = 0
Therefore, the function would go like this: (x - 3) * (x - 4) / (x +3) * (x + 4),
Operating the following we have:
(x ^ 2 -4 * x - 3 * x +12) / (x ^ 2 + 4 * x + 3 * x +12)
(x ^ 2 - 7 * x +12) / (x ^ 2 + 7 * x +12)
having asymptotes at x = -3 and x = -4, and intersection at (3.0) and (4.0), that is x = 3 and x = 4.
Horizontal asymptote.
Finally, in the case of the horizontal asymptote, which must be y = 10, what must be done is to divide all the terms by the highest degree term, in this case it is x ^ 2. It would be as follows:
(x ^ 2 / x ^ 2 - 7 * x / x ^ 2 + 12 / x ^ 2) / (x ^ 2 / x ^ 2 + 7 * x / x ^ 2 + 12 / x ^ 2)
Now, x ^ 2 / x ^ 2 = 1, and all the others have to be 0, therefore it would remain (1 + 0 + 0) / (1 + 0 + 0) = 1
So for the asymptote to be 10, the entire denominator must be multiplied by 10, thus being:
10 * (x ^ 2 - 7 * x +12) / (x ^ 2 + 7 * x +12)
(10x ^ 2 - 70 * x +120) / (x ^ 2 + 7 * x +12)
And this equation would already comply with everything required in the problem.
