Answer:
The answer is the option C
graph of \(3x\) minus \(3\), with discontinuity at negative \(2\), negative \(9\)
Step-by-step explanation:
we have
\(f(x)=\frac{9x^{2}+9x-18}{3x+6}\)
Simplify
\(f(x)=9\frac{(x^{2}+x-2)}{3(x+2)}\)
\(f(x)=3\frac{(x^{2}+x-2)}{(x+2)}\)
Step 1
Convert to a factored form the numerator
\(x^{2}+x-2=0\)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
\(x^{2}+x=2\)
Complete the square. Remember to balance the equation by adding the same constants to each side.
\(x^{2}+x+0.25=2+0.25\)
\(x^{2}+x+0.25=2.25\)
Rewrite as perfect squares
\((x+0.5)^{2}=2.25\)
Square root both sides
\(x+0.5=(+/-)1.5\)
\(x=-0.5(+/-)1.5\)
\(x=-0.5+1.5=1\)
\(x=-0.5-1.5=-2\)
so
\(x^{2}+x-2=(x-1)(x+2)\)
Step 2
Simplify the function f(x)
\(f(x)=3\frac{(x^{2}+x-2)}{(x+2)}=3\frac{(x-1)(x+2)}{(x+2)}\)
The domain of the function f(x) is all real numbers except the number \(x=-2\)
Because the denominator can not be zero
\(f(x)=3\frac{(x-1)(x+2)}{(x+2)}=3(x-1)=3x-3\)
\(f(x)=3x-3\) ------> with a discontinuity at \(x=-2\)
\(f(-2)=3(-2)-3=-9\)
The discontinuity is at point \((-2,-9)\)
the answer in the attached figure
