Answer:
y = (3x² + x - 2)/(x² - 1)
Step-by-step explanation:
(x - 1)(x + 1) = x² - 1
In the denominator
f(x)/(x² - 1)
f(x) has a factor (x + 1)
3 + (x + 1)/(x² - 1)
(3x² - 3 + x + 1)/(x² - 1)
y = (3x² + x - 2)/(x² - 1)
Answer:
[tex]\frac{3x(x+1)}{(x+1)(x-1)}[/tex]
Step-by-step explanation:
Vertical asymptotes are places where the denominator of a rational expression equals 0 because in those scenarios, we have something like k/0, where k is any expression. This is undefined.
Since x = 1 is a vertical asymptote, we need to make sure we put (x - 1) in the denominator because when we set that equal to 0, we get x = 1.
The domain says that it's all reals except 1 and -1. Domain is all the possible x values. We already covered 1 because we put (x - 1) in the denominator. Similarly, to make sure that the domain doesn't include -1, we need to put x + 1 in the denominator.
Now, we see that x = -1 is a hole. A hole is a removable point of discontinuity. This means that (x + 1) needs to be in the denominator and numerator. That way, we can "remove" it by cancelling it out from the top and bottom, but it still counts as a vertical asymptote.
Finally, there is a horizontal asymptote at y = 3. We get horizontal asymptotes when y approaches a value. A property of horizontal asymptotes is that when the degree of the numerator and denominator of a rational expression are equal, the horizontal asymptote is just the ratio of their leading coefficients. That's why I made sure that both the top and bottom had degrees of 2 by adding an x to the numerator, and I also added a 3 to the numerator so that the leading coefficient of the numerator is 3 and leading coefficient of the denominator is 1, making the ratio 3:1 = 3.
Hope this helps!
