How do you graph #f(x)=(3x^212x9)/(x^2+5x+4)# using holes, vertical and horizontal asymptotes, x and y intercepts?
see explanation.
First, factorise and simplify f(x)
with exclusion x ≠  1 which indicates a hole at x =  1
The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is nonzero for this value then it is a vertical asymptote.
Horizontal asymptotes occur as
divide terms on numerator/denominator by x
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To graph the function f(x)=(3x^212x9)/(x^2+5x+4), we can analyze its holes, vertical and horizontal asymptotes, as well as the x and y intercepts.

Holes: To find the holes, we need to factor the numerator and denominator. The numerator can be factored as 3(x+1)(x+3), and the denominator can be factored as (x+1)(x+4). We can cancel out the common factor (x+1), which results in f(x)=3(x+3)/(x+4). The hole occurs at x=1, as it was canceled out from the numerator and denominator.

Vertical Asymptotes: Vertical asymptotes occur where the denominator equals zero. By factoring the denominator, we find that it equals zero at x=1 and x=4. Therefore, the vertical asymptotes are x=1 and x=4.

Horizontal Asymptotes: To determine the horizontal asymptote, we compare the degrees of the numerator and denominator. Since the degree of the numerator (3x^212x9) is less than the degree of the denominator (x^2+5x+4), the horizontal asymptote is y=0.

Xintercepts: Xintercepts occur where the function crosses the xaxis. To find them, we set f(x) equal to zero and solve for x. In this case, we set 3(x+3)/(x+4) equal to zero, which gives us x=3. Therefore, the xintercept is (3, 0).

Yintercept: The yintercept occurs where the function crosses the yaxis. To find it, we set x equal to zero in the function f(x). Plugging in x=0, we get f(0)=3(0+3)/(0+4), which simplifies to f(0)=9/4. Therefore, the yintercept is (0, 9/4).
By considering these aspects, we can graph the function f(x)=(3x^212x9)/(x^2+5x+4) accurately.
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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.
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