Question

Which graph represents the function of f(x) = the quantity of 9 x squared plus 9 x minus 18, all over 3 x plus 6

A. graph of 3 x plus 6, with discontinuity at 2, 12

B. graph of 3 x plus 6, with discontinuity at negative 2, 0

C. graph of 3 x minus 3, with discontinuity at negative 2, negative 9

D. graph of 3 x minus 3, with discontinuity at 2, 3

Answer 1

Hello,

y=(9x²+9x-18)/(3x+6)=(9(x-1)(x+2))/(3(x+2))
If x≠-2 then y=3(x-1)==>y=3x-3
So we must exclude (-2,-9)

Answer C (graph n°3)

Answer 2

The answer is the option C

graph of
`3x` minus
`3`, with discontinuity at negative
`2`, negative
`9`

Explanation:

we have

`f(x)=(9x^(2)+9x-18)/(3x+6)`

Simplify

`f(x)=9((x^(2)+x-2))/(3(x+2))`

`f(x)=3((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((x^(2)+x-2))/((x+2))=3((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((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