How do you graph f(x)=(x^3−x)/(x^3+2x^2−3x) using holes, vertical and horizontal asymptotes, x and y intercepts?

Kendra Hudson 2022-09-13 Answered
How do you graph f ( x ) = x 3 - x x 3 + 2 x 2 - 3 x using holes, vertical and horizontal asymptotes, x and y intercepts?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

scrapbymarieix
Answered 2022-09-14 Author has 15 answers
You need to first factor to see if you can eliminate anything (this is when holes will occur).

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

There will be two holes: at x=0 and x=1. There will be vertical asymptotes at x=0, x=−3 and x=1 (since this is what makes the denominator 0 and hence undefined). However, the supposed vertical asymptote at x=1 and x=0 is in fact a hole.

The exact coordinates of the holes can be obtained by substituting x=a into the simplified function.

Hence, the holes will be at ( 0 , 1 3 ) and ( 1 , 1 2 ) .

For this function, there will be a horizontal asymptote at the ratio between the coefficents of the terms with highest degree in the numerator and denominator.

The horizontal asymptote is given by y = 1 1 = 1 .

As for intercepts, set the function to 0 and solve.

y intercept:
there are none, because both are eliminated when factoring (even though it does appear that there is a y-intercept on the graph, this is in fact a hole.

x-intercept:

You will find there is an x-intercept at x=−1.

The last thing that is requiblack to graph a rational function like this is end behavior. This can be found by picking a few numbers close to the asymptotes and checking their trend. For example, you can pick x=−3.5 and x=−3.001, and on the other side you can pick x=−2.999 and x=−2.5.

Doing this for all the vertical and horizontal asymptotes, you should get the following graph.
graph{(x^3 - x)/(x^3 + 2x^2 - 3x) [-10, 10, -5, 5]}
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-02-25
True or False. The graph of a rational function may intersect a horizontal asymptote.
asked 2021-06-23
Choose the correct term to complete each sentence. Limits of polynomial and rational functions can be found by _____, so long as the denominator of the rational function evaluated at c is not 0.
asked 2022-07-01
Prove that for any two distinct points of an irreducible curve there exists a rational function that is regular at both, and takes the value 0 at one and 1 at the other.
I think I can construct such a function, for example, u ( x , y ) = ( x a ) 2 + ( y b ) 2 for given two points ( a , b ) and ( c , d ). However, this doesn't work for general algebraically closed field, for example, the case of ( c , d ) = ( a + i , b + 1 ). Hence now I have no clue. Could you give me a hint for this problem?
asked 2022-09-03
Does the perimeter of a square vary inversely, directly or neither as the length of the side?
asked 2022-02-15
Let f be a rational function on a compact connected Riemann surface X. The rational function f induces a holomorphic map f:XR1(C).
Let x be a point on the Riemann sphere P1(C). ow can I check that if b is a branch point of f by looking at the derivative of f?
How does this work when X=P1(C)?
asked 2022-05-17
Q1: If you have a rational function, how do you know it will have oblique asymptote?
Q2: if you have a rational function, is long division the best way to find the oblique asymptote?
Any help, please
asked 2022-02-17
I am finding the divisor of f=(x1x0)1 on C, where C=V(x12+x22x02)PP2. Characteristic is not 2.
I am totally new to divisors. So the plan in my mind is first find an open subset U of C. In this case maybe it should be the complement of X=x2=0. Then I should look at ord(f) on this U. Then I get confused, since f is a rational function, how can f belong to k[U]?
Who knows?