How do you graph #f(x)=(2x^2+1)/(x^3-x)# using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

We can graph this equation using a sign chart.

First, let's identify the holes, vertical and horizontal asymptotes and x and y intercepts.

Horizontal Asymptotes
Observing the formula, we can see the degree of the denominator is higher than the numerator (The degree is the largest exponent in a polynomial). This means there will be an asymptote at #y=0#.

Vertical Asymptotes
In order to find vertical asymptotes, we muct factor the denominator. Factoring out an #x# from #x^3-x#, we will get #x(x^2-1)#. Equating both answers to 0 will allow us to find our asymptotes.

#x=0#

#x^2-1=0#
#x^2=1#
#sqrt(x^2)=sqrt(1)#
#x=+-1#
#x!=-1,0,1#

#x# can not equal any of these values as they all result in a 0 in the denominator. This will make the function undefined aka asymptotes.

Holes
Holes occur when there is a zero in both the numerator and denominator that will cancel. There are none in this formula.

X-Intercepts
To find x-intercepts, substitute 0 for #f(x)# and solve.

#0=(2x^2+1)/(x^3-x)#

There are no x-intercepts as the equation above would result in imaginary numbers as answers (#(isqrt(2))/sqrt(2),-(isqrt(2))/sqrt(2)#).

Y-Intercepts
To find y-intercepts substitute 0 for #x# and solve.

#y=(2(0)^(2)+1)/((0)^(3)-(0))#

As the denominator equals zero the equation is undefined. Therefore there are no y-intercepts.

Sign Chart
Using the values above, also known as critical values, we will create a sign chart. We do this by sorting the critical values from least to greatest, and surrounding them with #-prop# and #prop#.

We will then select a value for #x# in between the critical numbers.

We will test the #x# values for positivity by substituting them for #x# in the formula and write either the positive or negative symbol in the spaces we created. I'll do #x=10# as an example below.
#y=(2(10)^(2)+1)/((10)^(3)-(10))#
#y=67/330#

We don't necessarily care about the value, just whether it is positive or negative. Since it is positive, we will mark a #+# between #1# and #prop#. We will do this for all the values we have chosen.

We are now able to graph now that we know where the function is positive and negative and we have the asymptotes.
graph{(2x^2+1)/(x^3-x) [-10, 10, -5, 5]}
Notice how the graph is positive where we noted positive, and negative where we noted negative.

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Answer 2

To graph the function f(x) = (2x^2 + 1)/(x^3 - x), follow these steps:

  1. Determine the domain of the function by finding the values of x for which the denominator is equal to zero. In this case, set x^3 - x = 0 and solve for x. The solutions are x = 0, x = 1, and x = -1. Therefore, the domain of the function is all real numbers except x = 0, x = 1, and x = -1.

  2. Find the y-intercept by evaluating f(0). Substitute x = 0 into the function: f(0) = (2(0)^2 + 1)/(0^3 - 0) = 1/0. Since division by zero is undefined, there is no y-intercept.

  3. Determine the x-intercepts by solving the equation f(x) = 0. Set (2x^2 + 1)/(x^3 - x) = 0 and solve for x. The numerator can never be zero, so the x-intercepts occur when the denominator is zero. Set x^3 - x = 0 and factor out x: x(x^2 - 1) = 0. This gives x = 0, x = 1, and x = -1 as the x-intercepts.

  4. Identify any vertical asymptotes by finding the values of x for which the function approaches infinity or negative infinity. Vertical asymptotes occur when the denominator of the function approaches zero. In this case, x^3 - x = 0 at x = 0, x = 1, and x = -1. Therefore, there are vertical asymptotes at x = 0, x = 1, and x = -1.

  5. Determine any horizontal asymptotes by analyzing the behavior of the function as x approaches positive or negative infinity. Divide the leading terms of the numerator and denominator to find the horizontal asymptote. In this case, the leading terms are 2x^2 and x^3. Dividing them gives 2x^2/x^3 = 2/x. As x approaches positive or negative infinity, the function approaches 0. Therefore, there is a horizontal asymptote at y = 0.

  6. Identify any holes in the graph by canceling out common factors between the numerator and denominator. In this case, there are no common factors to cancel out, so there are no holes in the graph.

  7. Plot the x-intercepts, vertical asymptotes, and horizontal asymptote on the graph. Use additional points if necessary to sketch the curve.

Remember to label the axes and provide a title for the graph.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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