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