Question

How do you find domain and range for`sqrt(x^2-3x)` ?

Answer 1

Domain `(-oo , 0] uu [3 , +oo)`

Range `[0 , +oo)`

Explanation:

To stay in the realm of real numbers the expression inside the radical must be `>=0`

So we have the inequality: `x^2-3x>=0`

So first we equate the expression to `0`. This gives us the end points of the relevant intervals.

So:

`x^2-3x=0=> x(x-3)=0=>x=0 and x=3`.

Since the coefficient of the `x^2` in the expression is positive, the function has a minimum value. This means we are looking for values to the left of `0` and to the right of 3.

So our domain will be:

`(-oo , 0] uu [3 , +oo)`

Since minimum value of the expression is `0` and maximum value is:

`lim_(x->oo)(sqrt(x^2-3x))= oo`

The range is:

`[0 , +oo)`

Graph of `y = (sqrt(x^2-3x))`

graph{y= sqrt(x^2-3x) [-10, 10, -5, 5]}