Question

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

Short answer

The domain of the function \(y=\frac{\sqrt{x}}{5}\) is \([0, \infty)\).

Step-by-step solution

Understand the function

The function given is \(y=\frac{\sqrt{x}}{5}\). In order for this function to be defined and real, the value inside the square root must be nonnegative because it's not possible to take the square root of a negative number in the real number system. Hence, \(x\) must be greater than or equal to 0.

Write out the domain

The domain of this function is the set of all real numbers \(x\) that satisfy the condition \(x \geq 0\). In interval notation, this will be written as \([0, \infty)\).

Key concepts

Square Root Functions

When dealing with square root functions, we consider functions that include a square root of a variable, like \( y = \sqrt{x} \). A critical aspect to remember is that the radicand— the number or expression inside the square root sign—must be a nonnegative number to provide real number solutions. This means that for \( \sqrt{x} \), \( x \) itself must be zero or positive.

Square root functions have a characteristic shape known as the graph of \( y = \sqrt{x} \), which is a curve that starts at the origin (0,0) and rises gradually to the right. This graph reflects the fact that square root functions only yield real number results for nonnegative values of \( x \). Therefore, when asked to find the domain of a square root function, we must identify the set of all possible \( x \) values that do not lead to an imaginary or undefined number.

Interval Notation

To express the domain or range of functions systematically, mathematicians commonly use interval notation. This form of notation is an efficient way to communicate which numbers are included in a set without listing every single one. It uses brackets and parentheses to indicate the boundaries of the interval.

For example, the interval notation \( [0, \infty) \) includes all numbers from 0 to infinity, where the square bracket \( [ \) means 'including 0' and the parenthesis \( ) \) means 'up to but not including' infinity—since infinity is not a number but rather a concept representing an unbounded limit. This notation is crucial when defining the domain of functions like the square root because it shows at a glance the span of acceptable values for the function's variable.

Nonnegative Numbers

The nonnegative numbers are all the numbers that are not negative, including zero. This set is incredibly important in the domain of square root functions because only nonnegative numbers under a square root sign will result in real numbers after the square root is evaluated. That means the smallest possible value for \( x \) in our square root function is zero.

Understanding nonnegative numbers helps to comprehend why the domain of \( y = \frac{\sqrt{x}}{5} \) starts at zero. The term nonnegative includes all positive numbers and zero itself, so when considering domain restrictions prompted by a square root, we must include zero and all the positive numbers that follow, which in interval notation is expressed as \( [0, \infty) \).