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-x^{2}>=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].)