come up with a rational function that has

Answered question

2022-03-09

come up with a rational function that has four asymptotes and three x intercepts.

Answer & Explanation

RizerMix

RizerMix

Expert2023-04-23Added 656 answers

To create a rational function with four asymptotes and three x intercepts, we can start by using the formula:

f(x)=a(x-r1)(x-r2)(x-r3)(x-s1)(x-s2)(x-s3)(x-s4)

where a is a constant, r1,r2, and r3 are the x intercepts, and s1,s2,s3, and s4 are the asymptotes. To achieve the desired result, we need to choose appropriate values for these parameters.

Let's start by choosing the asymptotes. As the function has four asymptotes, we will use the equation of a hyperbola with vertical and horizontal asymptotes:

s1=2
s2=-2
s3=3i
s4=-3i

Next, we need to select the x-intercepts. To make sure that the function has exactly three x-intercepts, we will choose three distinct real numbers:

r1=-1
r2=0
r3=1

Finally, we need to determine the value of the constant a. We can do this by setting the leading coefficient of the function to 1. The leading coefficient is the coefficient of the highest-degree term in the numerator, which is a(x-r1)(x-r2)(x-r3). Since this term has degree 3, we need to choose a such that:

a=1b(r1)(r2)(r3)

where b is a constant that we can choose arbitrarily. Let's set b = 1 for simplicity.

Substituting all the values we have chosen, we get the final rational function:

f(x)=(x+1)x(x-1)8(x-2)(x+2)(x+3i)(x-3i)

This function has four vertical and horizontal asymptotes at x = 2, x = -2, x = 3i, and x = -3i, and three x-intercepts at x = -1, x = 0, and x = 1.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?