How do you graph #f(x)=(3x^2+2)/(x+1)# using holes, vertical and horizontal asymptotes, x and y intercepts?
V.A
H.A none
S.A
no
no holes

V.A are the zeros (undefined points) of the denominator
#x+1=0#
#x= 1#
vertical asymptotes are#x= 1# 
H.A if the degrees of the numerator is equal to the degree of denominator,
if the numerators degree > 1 + degree of denominator, there is a slant asymptote
degree of numerator is 2, degree of denominator is 1
#y=mx+b#
#(3x^2+2)/(x+1)#
#=3x+(3x+2)/(x+1)#
#=3x3+(5)/(x+1)#
#y=3x3#
slant asymptote is#y=3x3# 
#x# intercept is a point on the graph where#y=0#
#(3x^2+2)/(x+1)=0# no solution for#x inRR#
no#x# intercept 
#y# intercept is the point on the graph where#x=0#
#y=(3 * 0^2+2)/(0+1)#
#y=2#
#y# intercept is#(0,2)# 
no holes because the denominator doesn't cancel out
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To graph the function f(x) = (3x^2 + 2)/(x + 1), we can analyze its key features:

Holes: To find any holes in the graph, we need to check if there are any values of x that make the numerator and denominator equal to zero simultaneously. In this case, there are no such values, so there are no holes in the graph.

Vertical Asymptotes: Vertical asymptotes occur when the denominator of the function becomes zero. To find them, we set the denominator (x + 1) equal to zero and solve for x. In this case, x = 1 is the vertical asymptote.

Horizontal Asymptotes: To determine the horizontal asymptote(s), we examine the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. In this case, the degree of the numerator (2) is less than the degree of the denominator (1), so the horizontal asymptote is y = 0.

xintercepts: To find the xintercepts, we set the numerator (3x^2 + 2) equal to zero and solve for x. However, in this case, the numerator is never equal to zero, so there are no xintercepts.

yintercept: To find the yintercept, we substitute x = 0 into the function. Thus, the yintercept is (0, 2/1) or simply (0, 2).
By considering these features, we can plot the graph of f(x) = (3x^2 + 2)/(x + 1) accordingly.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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