SOLUTION: I am having trouble figuring out if the following equation has point discontinuity, vertical asymptote, or both. {{{ f(x)= x^2 - 10x + 24 / x + 9 }}} (I couldn't figure out how

Algebra ->  Rational-functions -> SOLUTION: I am having trouble figuring out if the following equation has point discontinuity, vertical asymptote, or both. {{{ f(x)= x^2 - 10x + 24 / x + 9 }}} (I couldn't figure out how       Log On


   

Question 6589: I am having trouble figuring out if the following equation has point discontinuity, vertical asymptote, or both. +f%28x%29=+x%5E2+-+10x+%2B+24+%2F+++x+%2B+9+
(I couldn't figure out how to edit it to show this to you, but it is not 24/x, rather it is x^2 - 10x + 24 ALL over x + 9.)
Thanks
Answer by ichudov(507) About Me  (Show Source):
You can put this solution on YOUR website!
You shoud write it as
+f%28x%29=+%28x%5E2+-+10x+%2B+24%29+%2F+++%28x+%2B+9%29+ (click on view source to see how I did it)
Find the roots of the numerator:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-10x%2B24+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-10%29%5E2-4%2A1%2A24=4.

Discriminant d=4 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--10%2B-sqrt%28+4+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-10%29%2Bsqrt%28+4+%29%29%2F2%5C1+=+6
x%5B2%5D+=+%28-%28-10%29-sqrt%28+4+%29%29%2F2%5C1+=+4

Quadratic expression 1x%5E2%2B-10x%2B24 can be factored:
1x%5E2%2B-10x%2B24+=+1%28x-6%29%2A%28x-4%29
Again, the answer is: 6, 4. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-10%2Ax%2B24+%29


They are in points 6, 4. Denominator has root x=-9. At that point -9, there is discontinuity similar to how 1/x behaves around 0.
When x approaches plus/minus infonity, this function approaches a straight line.

the graph is not great, there is a complete discontinuity around x=-9.