Question

What is the graph of the function f(x) = the quantity of x squared plus 5x plus 6, all over x plus 3?

Answer 1

positive sloping linear graph with hole at negative 3, negative 1 intersecting the x axis at negative two and the y axis at two

Answer 2

The graph of given function is shown below.

Explanation:

The given function is

`f(x)=(x^2+5x+6)/(x+3)`

Factorize the numerator.

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

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

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

Cancel out the common factor.

`f(x)=x+2`

It is a straight line because it is a one degree polynomial.

At x=0,

`f(0)=0+2=2`

The y-intercept of the function is (0,2).

Put f(x)=0

`0=x+2`

`x=-2`

The x-intercept of the function is (-2,0).

Equate the cancel factor equal to 0, to find the hole of the function.

`x+3=0\Rightarrow x=-3`

The function has hole at x=-3.

The graph of given function is shown below.