Suppose that f(x) and g(x) are polynomials in x. Can the graph...

Question

Suppose that f(x) and g(x) are polynomials in x. Can the graph of f(x)/g(x) have an asymptote if g(x) is never zero? Give reasons for your answer.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a student is studying the behavior of the function H of X, which is equal to the quantity of 3 X2 + 2X minus 5 in quantity, divided by the quantity of 2 X2 + 7 in quantity. They wonder if this function can have a horizontal asymptote, given that the denominator never equals zero. What should they conclude? We're given 4 possible choices as our answers. For choice A, we have yes, it can have a horizontal asymptote. For choice B, no, it cannot have any asymptotes. For choice C, we have yes, it can have a vertical asymptote. And for choice D, no, it can only have oblique asymptotes. So, to answer this question, we need to determine whether our function can have a horizontal asymptote. And we're actually going to look at all three types of asymptotes. We'll look at vertical asymptotes. We'll look at oblique asymptotes. Which are also called slanted. Asymptotes and we'll look at the horizontal asymptote, which is what we're looking for in this problem. So if we start by looking at vertical asymptotes, one of the conditions in order to have a vertical asymptote with a rational function is that our our denominator needs to be equal to 0 at some value of X. Now, in our case here, we see that our denominator is never equal to 0, therefore, we cannot have any vertical asymptotes. We won't have any vertical asymptotes with this function. Now, the condition to have an oblique asymptote is that you need to have the degree of the numerator to be 1 greater than the degree of the denominator. But if we look here, we see that the degree of our polynomial in the numerator is 2, and the degree of our polynomial in the denominator is also 2, so they're equal. So that means that we won't have any oblique asymptotes. Now, we will have a possibility of having a horizontal asymptote, since one of the conditions to have a horizontal asymptote is to have the degree of the polynomial in the numerator to be equal to the degree of the polynomial in the denominator. Since the present of presence of equal degrees in the numer denominator suggests that as X approaches infinity, the function approaches a constant value, which would indicate a horizontal asymptote. So since we can't have a horizontal asptote, that means that the correct answer for this problem is answer choice A. Yes, it can have a horizontal asymptote. So I hope that this explanation has been useful, and I'll see you in the next video.

Suppose that f(x) and g(x) are polynomials in x. Can the graph of f(x)/g(x) have an asymptote if g(x) is never zero? Give reasons for your answer.

Key Concepts

Polynomials

Asymptotes

Rational Functions

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