Answer:
The answer is the option C
graph of
minus
, with discontinuity at negative
, negative ![9](https://tex.z-dn.net/?f=9)
Step-by-step explanation:
we have
![f(x)=\frac{9x^{2}+9x-18}{3x+6}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B9x%5E%7B2%7D%2B9x-18%7D%7B3x%2B6%7D)
Simplify
![f(x)=9\frac{(x^{2}+x-2)}{3(x+2)}](https://tex.z-dn.net/?f=f%28x%29%3D9%5Cfrac%7B%28x%5E%7B2%7D%2Bx-2%29%7D%7B3%28x%2B2%29%7D)
![f(x)=3\frac{(x^{2}+x-2)}{(x+2)}](https://tex.z-dn.net/?f=f%28x%29%3D3%5Cfrac%7B%28x%5E%7B2%7D%2Bx-2%29%7D%7B%28x%2B2%29%7D)
Step 1
Convert to a factored form the numerator
Group terms that contain the same variable, and move the constant to the opposite side of the equation
![x^{2}+x=2](https://tex.z-dn.net/?f=x%5E%7B2%7D%2Bx%3D2)
Complete the square. Remember to balance the equation by adding the same constants to each side.
![x^{2}+x+0.25=2+0.25](https://tex.z-dn.net/?f=x%5E%7B2%7D%2Bx%2B0.25%3D2%2B0.25)
![x^{2}+x+0.25=2.25](https://tex.z-dn.net/?f=x%5E%7B2%7D%2Bx%2B0.25%3D2.25)
Rewrite as perfect squares
![(x+0.5)^{2}=2.25](https://tex.z-dn.net/?f=%28x%2B0.5%29%5E%7B2%7D%3D2.25)
Square root both sides
![x+0.5=(+/-)1.5](https://tex.z-dn.net/?f=x%2B0.5%3D%28%2B%2F-%291.5)
![x=-0.5(+/-)1.5](https://tex.z-dn.net/?f=x%3D-0.5%28%2B%2F-%291.5)
![x=-0.5+1.5=1](https://tex.z-dn.net/?f=x%3D-0.5%2B1.5%3D1)
![x=-0.5-1.5=-2](https://tex.z-dn.net/?f=x%3D-0.5-1.5%3D-2)
so
Step 2
Simplify the function f(x)
![f(x)=3\frac{(x^{2}+x-2)}{(x+2)}=3\frac{(x-1)(x+2)}{(x+2)}](https://tex.z-dn.net/?f=f%28x%29%3D3%5Cfrac%7B%28x%5E%7B2%7D%2Bx-2%29%7D%7B%28x%2B2%29%7D%3D3%5Cfrac%7B%28x-1%29%28x%2B2%29%7D%7B%28x%2B2%29%7D)
The domain of the function f(x) is all real numbers except the number ![x=-2](https://tex.z-dn.net/?f=x%3D-2)
Because the denominator can not be zero
------> with a discontinuity at ![x=-2](https://tex.z-dn.net/?f=x%3D-2)
![f(-2)=3(-2)-3=-9](https://tex.z-dn.net/?f=f%28-2%29%3D3%28-2%29-3%3D-9)
The discontinuity is at point ![(-2,-9)](https://tex.z-dn.net/?f=%28-2%2C-9%29)
the answer in the attached figure