Question

Find the domain of the function. $$y=8 \sqrt{\frac{5}{2} x}$$

Short answer

The domain of the function \(y = 8 \sqrt{\frac{5}{2}x}\) is \(x \geq 0\).

Step-by-step solution

Understand the condition for the square root

The expression under square root, \(\frac{5}{2}x\), must be non-negative because the square root of negative numbers is undefined in real number set. Therefore, we have to solve this inequality \(\frac{5}{2}x \geq 0\) to find the domain.

Solve the inequality

We can solve the inequality by isolating \(x\). In order to do this, we can multiply both sides of the inequality by \(\frac{2}{5}\) (the reciprocal of \(\frac{5}{2}\)). So, we get \(x \geq 0\) after simplifying.

Write down the answer

The solution of the inequality describes the domain of the function. So, the domain of the function \(y = 8 \sqrt{\frac{5}{2}x}\) is \(x \geq 0\). This includes all real numbers greater than or equal to zero.

Key concepts

Square Root in Functions

Working with functions that include square roots can be a bit tricky, especially when trying to determine their domain. The key thing to remember is that, in the context of real numbers, the quantity under the square root, known as the radicand, must be non-negative. This is because the square roots of negative numbers aren't real; they are complex numbers, which are beyond the scope of most high school mathematics courses.

To ensure the radicand is non-negative, we set up an inequality that represents this condition. For example, in the function \(y = 8 \sqrt{\frac{5}{2} x}\), we have to examine the expression \(\frac{5}{2}x\) and solve the inequality \(\frac{5}{2}x \geq 0\). Solving this will provide us with the values of \(x\) that make the function real and therefore define the domain. It's important to practice these steps to become proficient in determining the domain of square root functions.

Solving Inequalities

Inequalities tell us about the relative size of two values and solving them is a critical skill in mathematics, particularly when finding the domain of a function. When you encounter an inequality like \(\frac{5}{2}x \geq 0\), it represents a range of possible values for \(x\) that satisfy the condition.

To solve such an inequality, apply the same principles as you would for an equation: isolate the variable you're solving for, taking care to reverse the inequality sign when multiplying or dividing by a negative number. In our example, since \(\frac{5}{2}\) is positive, we can safely multiply both sides by its reciprocal, \(\frac{2}{5}\), without reversing the inequality, giving us \(x \geq 0\). Remember, solving inequalities carefully is crucial to correctly identifying the function's domain.

Domain of Real Number Functions

The domain of a real number function is the set of all possible values of the independent variable (usually \(x\)) for which the function produces a real number output. When examining the domain of functions like \(y = 8 \sqrt{\frac{5}{2} x}\), it's essential to identify any restrictions that would result in non-real numbers.

For instance, since we can't have square roots of negative numbers in real functions, we establish that \(x\) must be greater than or equal to zero. This is how we deduced from the previous steps that the domain of the function is all real numbers \(\geq 0\). It's always useful to sketch a number line to visualize the domain clearly. Understanding the domain is fundamental for graphing functions and for understanding their behavior throughout their entire range of possible input values.