Find the domain of $y=\sqrt{3 x-8}$.
To find the domain of the function \(y = \sqrt{3x - 8}\), we need to determine the set of all real values of \(x\) for which the function is defined. In the set of real numbers, the square root of a number is defined if and only if the expression inside the square root (the radicand) is greater than or equal to zero. Thus, we must have the radicand \(3x - 8 \geq 0\).
To isolate the variable \(x\), we first add 8 to both sides of the inequality:
\[3x - 8 + 8 \geq 0 + 8\]
\[3x \geq 8\]
Next, we divide both sides by 3:
\[x \geq \frac{8}{3}\]
This gives us the range of values for \(x\) that satisfy the condition for the square root function to be defined.
Based on our calculation, the function \(y = \sqrt{3x - 8}\) is defined for all \(x\) such that \(x \geq \frac{8}{3}\). Comparing this result with the provided options, it matches the first option. Therefore, the domain of the function is \(x \geq \frac{8}{3}\).
The concept of domain is essential in many fields. For example, if you are designing a computer program that calculates the square root of a user's input, the program must ensure that it only receives non-negative numbers to avoid errors. By defining the domain as \(x \geq \frac{8}{3}\), you establish the valid inputs that the program can process safely, preventing crashes or invalid outputs. This is a fundamental aspect of robust software development and mathematical modeling.