Fill in the blank(s) to correctly complete each sentence. The function $g\left(x\right)=\text{√}x$ has domain ________.

Verified step by step guidance

1. Recall that the function \( g(x) = \sqrt{x} \) involves a square root, and the expression inside the square root (called the radicand) must be greater than or equal to zero for the function to be defined in the real numbers.
2. Set the radicand \( x \) to be greater than or equal to zero: \( x \geq 0 \). This ensures that the square root is defined for all \( x \) values that are zero or positive.
3. Express the domain of \( g(x) \) as all real numbers \( x \) such that \( x \geq 0 \). In interval notation, this is written as \( [0, \infty) \).
4. Therefore, the domain of \( g(x) = \sqrt{x} \) is all real numbers \( x \) where \( x \geq 0 \), or equivalently, \( [0, \infty) \).
5. This means you can input any number zero or greater into the function \( g(x) \) and get a real number output.

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Determining the domain involves identifying values that do not cause undefined expressions, such as division by zero or taking the square root of negative numbers.

Square Root Function Properties

The square root function, √x, is only defined for non-negative real numbers because the square root of a negative number is not a real number. Therefore, the input x must be greater than or equal to zero to keep the function real-valued.

Inequalities and Set Notation

To express the domain, inequalities are used to specify the range of valid x-values, such as x ≥ 0. Set notation or interval notation (e.g., [0, ∞)) is commonly used to represent the domain clearly and concisely.