Find the Domain of the Square Root of a Quadratic Function

Find the domain of a square root function whose radicand is a quadratic expression follwoing a step by step process. As many examples as needed to learn the steps may be generated. There is a graph at the bottom of the page that helps you further understand graphically the solution to the question shown below. You may want to first review solving quadratic inequalities on which is based the present worksheet.





Step by step solution

STEP 1: The radicand (the expression under the radical) must be positive or equal to 0 so that $f(x)$ takes real values. Write this condition as an inequality.

STEP 2: Solve the inequality obtained in step (1). You may use any method but the method of the discriminant (solving quadratic inequalities) is used here because it works in all situations.

STEP 3: Write the domain using interval notation.

Graphical Meaning of Solution



Below are shown the graphs of the given function (blue) and its radicand (green) which is a quadratic expression and its graph is a parabola. The graph of the function exist only for values of $x$ for which the radicand is positive (green graph above $x$-axis). (Change scales if necessary)