Question 726020
The domain appears to be all values of x except x = -1 and x = 0
The range appears to be all values of y.
The zeroes (roots) appear to be x = -2/3 and x = 1.


you find the domain by looking for all values of x that will result in a value of y.


all values of x will result in a value of y except when x = -1 and when x = 0.


when x = -1 or when x = 0, the denominator of the original equation will be equal to 0 which makes the value of y undefined.


therefore x = -1 and x = 0 can't be in the domain.


since there are no further restrictions as to what the value of x can be, then the domain is all values of x except at x = -1 and x = 0.


the zeroes of the equation are found by setting the equation equal to 0 and solving for x.

it's best to simplify the equation first.


start with (3x^3 - x^2 - 2x) / (x * (x+1)^2


since (3x^3 - x^2 - 2x) / x is equal to (3x^2 - x - 2), the equation becomes:


(3x^2 - x - 2) / (x+1)^2


set this equal to 0 to get:


(3x^2 - x - 2) / (x + 1)^2 = 0


multiply both sides of the equation by (x + 1)^2 to get:


(3x^2 - x - 2) = 0


factor to get:


(3x + 2) (x - 1) = 0


solve for x to get:


x = -2/3 or x = 1


these are the roots of the quadratic equation which is the point at which the equation crosses the x-axis which is the zeroes of the equation.


you can confirm that the solutions are correct by graphing the equation.


the graph of the equation looks like this:


{{{graph(600,600,-10,10,-5,15,(3x^3-x^2-2x)/(x*(x+1)^2))}}}


from this graph, you can see that the zeroes of the graph are around x = -2/3 and x = 1.


from this graph, you can see that there appears to be an asymptote at around x = -1 although it's not real easy to see that.


from this graph, you cannot see that there is a hole at x = 0


the value of y is undefined at x = -1 and also undefined at x = 0.


this is because the denominator of the equation is 0 at those values of x.


at x = -1, this results in an asymptote.


at x = 0, this results in a hole.


if you try to find the value of y when x = 0 and when x = -1, you will not be able to.


you can find a value of y for any other value of x other than 0 or -1.