How do you graph and find the discontinuities of #F(x) = (-2x^2 + 1)/(2x^3 + 4x^2)#?

Answer 1

Solve for the asymptotic discontinuities

To make it easier to look for the discontinuities, it helps to completely factor all the expressions first. #(-2x^2+1)/(2x^3+4x^2)# #=(-2x^2+1)/(2x^2(x+2)#
The discontinuities of a rational function are asymptotic discontinuities. These are the values of of #x# that will cause the denominator to be 0 (making it undefined). To solve for the asymptotic discontinuities, equate the denominator to 0 and find the solutions. #2x^2(x+2)=0#

We can separate this into two equations:

Equation 1: #2x^2=0# #x=0#
Equation 2: #x+2=0# #x+2-2=0-2# #x=-2#
The asymptotes are #x=0# and #x=-2#. Graph these two lines on your paper using a dotted line.

As for graphing, you can do that by creating a table of values.

It should end up like this: graph{(-2x^2+1)/(2x^3+4x^2) [-10, 10, -5, 5]}

You will notice that the graph looks like it's approaching x=0 and x=-2, but it will never actually touch it.

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Answer 2

To graph and find the discontinuities of the function F(x) = (-2x^2 + 1)/(2x^3 + 4x^2), follow these steps:

  1. Determine the domain of the function by identifying any values of x that would make the denominator equal to zero. In this case, set the denominator equal to zero and solve for x: 2x^3 + 4x^2 = 0 Factor out common terms: 2x^2(x + 2) = 0 Set each factor equal to zero and solve for x: 2x^2 = 0 or x + 2 = 0 x = 0 or x = -2

  2. The domain of the function is all real numbers except for x = 0 and x = -2.

  3. To graph the function, plot points by substituting various x-values into the equation and calculating the corresponding y-values. You can choose values such as -3, -2, -1, 0, 1, 2, and 3 to get a sense of the shape of the graph.

  4. Determine the behavior of the function as x approaches the values where the function is not defined (discontinuities). In this case, as x approaches 0 or -2, the function approaches positive or negative infinity, respectively.

  5. Plot the points obtained from step 3 on a graph and connect them smoothly to form the graph of the function, taking into account the behavior near the discontinuities.

Note: It is recommended to use graphing software or a graphing calculator to visualize the graph accurately.

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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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