In Exercises 61–64, find the domain of each function. f(x) = √(2x^2 - 5x + 2)

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1. Identify the expression inside the square root: 2x^2 - 5x + 2.
2. Set the expression inside the square root greater than or equal to zero because the square root of a negative number is not defined in the set of real numbers: 2x^2 - 5x + 2 \geq 0.
3. Factorize the quadratic equation 2x^2 - 5x + 2. Look for two numbers that multiply to 2*2=4 and add up to -5. These numbers are -4 and -1, so factorize as (2x - 1)(x - 2).
4. Set each factor greater than or equal to zero and solve for x: 2x - 1 \geq 0 and x - 2 \geq 0.
5. Combine the solutions from the inequalities to find the intervals where the original expression is non-negative. This will give the domain of the function.

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For real-valued functions, this often involves identifying values that do not lead to undefined expressions, such as division by zero or taking the square root of a negative number.

Square Root Function

A square root function, denoted as √(x), is defined only for non-negative values of x. This means that the expression inside the square root must be greater than or equal to zero to yield real number outputs. Understanding this is crucial for determining the domain of functions involving square roots.

Quadratic Inequalities

Quadratic inequalities involve expressions of the form ax^2 + bx + c ≥ 0 or similar forms. To find the domain of the function f(x) = √(2x^2 - 5x + 2), one must solve the inequality 2x^2 - 5x + 2 ≥ 0, which requires finding the roots of the quadratic and analyzing the intervals to determine where the expression is non-negative.