What is the:
- vertical asymptote
-horizontal asymptote
-non permissible value
of y=(x^2+x-12)/(x+4) ?
So this simplifies to a linear function y=x-3. BUT the oblique asymptote is also y=x-3. So does this graph not exist? I am really confused.
in the simplification process you would have to state it this way ....
y=(x^2+x-12)/(x+4)
= (x-3)(x+4)/(x+4)
= x - 3 , x ≠ -4
So in effect
y=(x^2+x-12)/(x+4)
is the same as
y = x - 3 for every value of x except -4
If we sub x = -4 into y = x-3, we get y = -7
If we sub x = -4 into y=(x^2+x-12)/(x+4) , we get 0/0
So the function y=(x^2+x-12)/(x+4)
is the same as y = x-3, with a hole at (-4,-7)
with no vertical or horizontal asymptotes
Verification:
http://www.wolframalpha.com/input/?i=y%3D%28x%5E2%2Bx-12%29%2F%28x%2B4%29
Why do we have to sub it back into the original function to get 0/0? If we got 0/x then it means it's a vertical asymptote right?
and what about the oblique asymptote? We don't worry about it then?
To determine the vertical asymptote, horizontal asymptote, and non-permissible values of the function y = (x^2 + x - 12)/(x + 4), we can follow these steps:
Step 1: Determine the vertical asymptote(s):
To find the vertical asymptote(s) of the function, we need to determine the values of x for which the denominator of the rational function is equal to zero. In this case, the denominator is (x + 4), so we set it equal to zero and solve for x:
x + 4 = 0
x = -4
Therefore, the vertical asymptote of the function is x = -4.
Step 2: Determine the horizontal asymptote:
To find the horizontal asymptote of the function, we examine the degree of the numerator (1st degree) and the denominator (1st degree) of the function. Since both degrees are the same, we divide the leading coefficients of the numerator and denominator to find the horizontal asymptote. In this case, the leading coefficients are both 1.
Therefore, the horizontal asymptote of the function is y = 1.
Step 3: Identify the non-permissible value(s):
Non-permissible values refer to the values of x for which the function is undefined. In rational functions, the non-permissible value(s) occur when the denominator is equal to zero. In this case, the non-permissible value is x = -4, since it makes the denominator (x + 4) equal to zero.
So, to summarize:
- The vertical asymptote of the function y = (x^2 + x - 12)/(x + 4) is x = -4.
- The horizontal asymptote of the function is y = 1.
- The non-permissible value of the function is x = -4.
Regarding your confusion about the oblique asymptote, it seems there was a mistake in your simplification. The function y = (x^2 + x - 12)/(x + 4) does not simplify to y = x - 3. Instead, it simplifies to y = x - 5 with a horizontal asymptote of y = x - 5. So, the graph of the function does exist, and the oblique asymptote is y = x - 5.