Determining the Domain of a Function Involving a Square Root: f(x) √(15x^2 - x - 2)

When dealing with functions involving square roots, it is essential to determine the domain - the set of all possible input values for which the function is defined. Specifically, the expression inside the square root must be non-negative. In this article, we will walk through how to find the domain of the function f(x) √(15x^2 - x - 2).

Introduction to the Problem

The given function is f(x) √(15x^2 - x - 2). Since the square root function is only defined for non-negative values, the expression inside the square root must be greater than or equal to zero. Therefore, we need to solve the inequality 15x^2 - x - 2 ≥ 0 to determine the domain of the function.

Finding the Roots of the Quadratic Equation

To solve the inequality, we first need to find the roots of the corresponding quadratic equation:

15x^2 - x - 2 0

The solution to a quadratic equation can be found using the quadratic formula:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

For the equation 15x^2 - x - 2 0, the coefficients are:

a 15

b -1

c -2

Calculate the discriminant:

b^2 - 4ac (-1)^2 - 4 cdot 15 cdot (-2) 1 120 121

Since the discriminant is positive, there are two distinct real roots. Calculate the roots:

x frac{-(-1) pm sqrt{121}}{2 cdot 15} frac{1 pm 11}{30}

Calculate the two roots:

x_1 frac{1 11}{30} frac{12}{30} frac{2}{5}

x_2 frac{1 - 11}{30} frac{-10}{30} -frac{1}{3}

Testing the Intervals

Having found the roots, we can determine the intervals where the quadratic expression is non-negative. The intervals to test are:

-infty x leq -frac{1}{3}

-frac{1}{3} leq x leq frac{2}{5}

x geq frac{2}{5}

Choose a test point from each interval:

x -1 in -infty x leq -frac{1}{3}

x 0 in -frac{1}{3} leq x leq frac{2}{5}

x 1 in x geq frac{2}{5}

Substitute these test points into the quadratic expression 15x^2 - x - 2 to verify non-negativity:

15(-1)^2 - (-1) - 2 15 1 - 2 14 0

15(0)^2 - 0 - 2 -2 0

15(1)^2 - (1) - 2 15 - 1 - 2 12 0

Based on this testing, we conclude that the quadratic expression is non-negative in the intervals -infty leq x leq -frac{1}{3} and frac{2}{5} leq x.

Conclusion

The domain of the function f(x) √(15x^2 - x - 2) is:

boxed{-infty leq x leq -frac{1}{3} cup frac{2}{5} leq x}

By following a systematic approach involving the quadratic formula and interval testing, we can effectively determine the domain of any function involving square roots.