- Identify the domain condition for the square root function: \(3 - 5y \geq 0\).
- Solve the linear inequality for \(y\): \(3 \geq 5y\) becomes \(y \leq \frac{3}{5}\).
- Match the result to the given options: \(y \leq \frac{3}{5}\) corresponds to option D.
- Final answer: \(\boxed{D}\)
Explanation
- Understanding the Domain Condition
The domain of a function refers to the set of all possible input values (in this case, \(y\)) for which the function produces a real output. For the square root function \(f(y) = \sqrt{3 - 5y}\), the expression inside the square root must be non-negative because the square root of a negative number is not a real number. Therefore, we must have \(3 - 5y \geq 0\).
- Solving the Inequality
To find the range of values for \(y\), we solve the inequality \(3 - 5y \geq 0\).
- Subtract \(3\) from both sides of the inequality: \(-5y \geq -3\).
- Divide both sides by \(-5\). When dividing or multiplying both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. Thus, \(y \leq \frac{-3}{-5}\).
- Simplifying the fraction gives \(y \leq \frac{3}{5}\).
- Conclusion
Comparing our result \(y \leq \frac{3}{5}\) to the provided options, we can see that it matches option D. Thus, the set of all real numbers in the domain of the function is \(y \leq \frac{3}{5}\). The final answer is \(\boxed{D}\).
Examples
The concept of a function domain is vital in fields like engineering and economics. For instance, if you are modeling the cost of a manufacturing process as a function of time, the domain represents the valid time intervals during which the process can operate. Similarly, if a health metric is calculated as the square root of a nutrient intake value, the domain ensures you only consider realistic intake amounts that avoid mathematically undefined results. Understanding these boundaries helps prevent errors and ensures models remain valid.