SOLUTION: Given that f(x)=x^2-1 divided by x. Find the following
a. V.A
b. H.A
c. X-intercepts
d. y-intercepts
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Question 134743: Given that f(x)=x^2-1 divided by x. Find the following
a. V.A
b. H.A
c. X-intercepts
d. y-intercepts
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Vertical asymptotes are located at where f(a) would be undefined because the rational function would have a zero denominator. In this case, f(0) is undefined because of the x in the denominator, hence there is a vertical asymptote at .
Horizontal asymptotes do not exist when the degree of the numerator polynomial is larger than the degree of the denominator polynomial. In this case the degree of the numerator polynomial is 2 and the degree of the denominator is 1. Therefore, there is no horizontal asymptote.
There is, however, a slant asymptote. When the degree of the numerator is larger than the degree of the denominator, perform polynomial long division or synthetic division of the denominator into the numerator. The quotient, excluding any remainder, is an equation of the slant asymptote. In this case, divided by gives as a quotient excluding the remainder, therefore the equation of the slant asymptote is .
x-intercepts are ordered pairs of the form (p,0) such that f(p)=0, in other words, where a graph of the function intersects the x-axis. Since if and only if and , we only need to set the numerator equal to zero and solve. Then, if necessary, exclude any roots that are not in the domain of the original function, namely those values discovered in our investigation of the vertical asymptotes. For this case, solve:
, so or
Therefore the x-intercepts are (1,0) and (-1,0).
y-intercepts are where the function intersects the y-axis, so it is a point (0,f(0)). In this case, the function is not defined at f(0), so there is no y-intercept.
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