# How to find domain for #f(x) = sqrt(x^2 - 5x)#?

##### 1 Answer

(-infinity,0] U [5,infinity)

#### Explanation:

I often find it easiest to see what the limiting factor is before I start with domain problems.

Can you square any number from negative to positive infinity? Yes.

Can you multiply any number from negative to positive infinity? Yes.

Can you square root any number from negative to positive infinity? No.

Recall that you cannot take the square root of a negative number, so this means that everything under the radical must *always* be positive. Now we have the beginnings of a domain:

Factor and find the zeros of the quadratic. Note: Do NOT divide by x! You will end up losing a solution! The factored form of the equation should look like

Now, it is necessary to test for whether the function is positive or negative at these values. You know that since the graph is a parabola, it should pass through the x axis at most twice (as it does here).

graph{x^2-5x [-10, 10, -10, 10]}

When a graph passes over the x axis, its outputs change sign, positive to negative or vice versa. Here, I would advise setting up a number line to test for values before x=0, between x=0 and x=5, and after x=5. Specific numbers aren't needed, you just need to know if it is positive or negative.

Before x=0: you can use some very big negative number, say -100.

Next you have the area between x=0 and x=5. Plug in a number, say 3.

Finally numbers greater than x=5. Put in 100.

Combining all the domains and you get (-infinity,0] U [5,infinity) brackets are used for the 0 and 5 because they also lead to non negative numbers within the radical. The U symbol means union, and it just combines two domains or sets.