### Finding the domain of a function with a square root

Find the domain of the function .

Solution

In order for the square root to make sense, only x-values which make the expression under the root sign non-negative can be used.

So in order to take the square root, we must have 2x-x2>=0.

Factoring, we obtain the inequality x(2-x)>=0. There are a number of ways to solve this inequality. To review solving polynomial inequalities, see the examples from the inequality section.

In this example, we will use a number line method, sometimes called a "sign chart," to solve the inequality. First, we find the roots of each factor:

 x=0 ==> x=0 2-x=0 ==> x=2

We can think of the roots "cutting off" intervals of the real number line: x < 0; 0 < x < 2; and x > 2. Then we determine the sign of each factor of the quadratic (in this case, x and 2-x) on each of the intervals.

 0 2 _______________________________________> x - - - + + + + + + 2-x + + + + + + - - - x(2-x) - - - + + + - - -

Let's look at the factor 2-x. The table indicates that the sign of 2-x is: positive (+++) when x < 0; positive again (+++) when 0 < x < 2; and negative (---) when x > 2.

Similarly, we see that the factor x is negative when x < 0; positive when 0 < x < 2; and positive when x > 2.

So therefore, the product x(2-x) is negative (---) when x < 0; positive when 0 < x < 2; and negative when x > 2.

Since we are looking for where x(2-x) >= 0, this means that the solution set is 0 <= x <= 2. (in interval notation, this is: [0,2].)