Find the domain of each function. $f(x) = \sqrt{2x^2 - 5x + 2}$

Verified step by step guidance

1. Identify the function given: \(f(x) = \sqrt{2x^2 - 5x + 2}\). Since this is a square root function, the expression inside the square root must be greater than or equal to zero for the function to be defined.
2. Set up the inequality for the radicand (the expression inside the square root): \$2x^2 - 5x + 2 \geq 0$.
3. Solve the quadratic inequality by first finding the roots of the quadratic equation \$2x^2 - 5x + 2 = 0\(. Use the quadratic formula: \)x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\(, where \)a=2\(, \)b=-5\(, and \)c=2$.
4. Once the roots are found, determine the intervals on the number line where the quadratic expression \$2x^2 - 5x + 2$ is greater than or equal to zero by testing values in each interval.
5. Write the domain of \(f(x)\) as the union of intervals where the inequality holds true, since these are the values of \(x\) for which the function is defined.

Key Concepts

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For functions involving square roots, the expression inside the root must be non-negative to produce real outputs. Identifying the domain ensures the function's outputs are real numbers.

Inequalities Involving Quadratic Expressions

To find where a quadratic expression is non-negative, solve the inequality by first finding the roots of the quadratic equation. Then, determine intervals where the quadratic is positive or zero by testing values or using the parabola's shape. This helps identify valid x-values for the function.

Factoring Quadratic Expressions

Factoring a quadratic expression involves rewriting it as a product of two binomials. This step is crucial to find the roots of the quadratic equation easily. For example, factoring 2x² - 5x + 2 helps locate critical points that define the domain boundaries.