How do you graph #f(x)= (x^3+1)/(x^2-4)#?

Answer 1

Graph of #y=(x^3+1)/(x^2-4)#
graph{(x^3+1)/(x^2-4) [-40, 40, -20,20]}

There is no secret to graph a function. # # Make a table of value of #f(x)# and place points. To be more accurate, take a smaller gap between two values of #x#

Better, combine with a sign table, and/or make a variation table of f(x). (depending on your level)

# # # # Before to start to draw, we can observe some things on #f(x)# Key point of #f(x)#: # # # #
Take a look to the denominator of the rational function : #x^2-4#
Remember, the denominator can't be equal to #0#

Then we will be able to draw the graph, when :

#x^2-4!=0 <=> (x-2)*(x+2)!=0 <=> x!=2# & #x!=-2#
We name the two straight lines #x=2# and #x=-2#, vertical asymptotes of #f(x)#, ie, that the curve of #f(x)# never crosses this lines. # #
Root of #f(x)# :
#f(x)=0 <=> x^3+1=0<=>x=-1#
Then :#(-1,0) in C_f#
Note : #C_f# is the representative curve of #f(x)# on the graph # # # # # #

N.B : J'ai hésité à te répondre en français, mais comme nous sommes sur un site anglophone, je prefère rester dans la langue de Shakespeare ;) Si tu as une question n'hésite pas!

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Answer 2

To graph the function f(x) = (x^3 + 1)/(x^2 - 4), follow these steps:

  1. Determine the domain of the function by finding the values of x for which the denominator (x^2 - 4) is equal to zero. In this case, x cannot be equal to 2 or -2, as it would make the denominator zero.

  2. Identify any vertical asymptotes by finding the values of x that make the denominator zero. In this case, x = 2 and x = -2 are vertical asymptotes.

  3. Determine any horizontal asymptotes by analyzing the degrees of the numerator and denominator. Since the degree of the numerator (3) is greater than the degree of the denominator (2), there is no horizontal asymptote.

  4. Find the x-intercepts by setting the numerator (x^3 + 1) equal to zero and solving for x. In this case, there are no x-intercepts.

  5. Find the y-intercept by substituting x = 0 into the function. In this case, the y-intercept is (0, 0.25).

  6. Plot additional points by selecting various x-values and calculating the corresponding y-values using the function.

  7. Draw the graph, connecting the plotted points and taking into account the vertical asymptotes.

Note: It may be helpful to use a graphing calculator or software 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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