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When we talk about the domain of a function, we are referring to all the possible inputs or x-values that the function can accept without resulting in undefined expressions. In the relation \( y = \sqrt{x + 3} \), we need to determine the values of \( x \) for which the square root expression is defined. Since the square root function is undefined for negative numbers, the expression \( x + 3 \) must be zero or positive.
Solving the inequality \( x + 3 \geq 0 \), we find that \( x \geq -3 \). Thus, the domain is all real numbers greater than or equal to -3.

• The domain is \( x \geq -3 \).
• It ensures the square root remains real-valued.

The range of a function is concerned with all possible outputs or y-values the function can produce. Due to the characteristics of a square root, which only outputs non-negative values, the range for this function will be \( y \geq 0 \). Hence, for every x-value in the domain, the function returns a non-negative y-value, complying with the square root's definition.

The vertical line test is a simple visual method used to determine whether a relation is a function. It involves drawing vertical lines through the graph of the relation. If any vertical line crosses the graph at more than one point, the relation is not a function, because a single x-value would correspond to multiple y-values.

In the case of \( y = \sqrt{x + 3} \), when plotted, the graph represents only one y-value for every x-value in the domain. This is indicative of a function. A vertical line can cross the graph at most once for each x in the domain:

• Every x-value matches precisely one y-value.
• Square root functions only output one value for any given input.

This means that the relation satisfies the vertical line test criteria, confirming that \( y = \sqrt{x + 3} \) is indeed a function.

The square root function is an important type of function that often arises in mathematical problems. The basic square root function is \( y = \sqrt{x} \) but it can be altered with constant shifts or stretches as seen in \( y = \sqrt{x + 3} \). Understanding how modifications alter the graph's position is crucial.

• Adding a number inside the root like \( x + c \) shifts the graph horizontally.
• In \( y = \sqrt{x + 3} \), the graph shifts 3 units to the left, starting at \( x = -3 \).

Square root functions always return non-negative results as they represent a principal square root. This characteristic affects the function's domain and range. In the given problem, understanding these shifts and domains is key to interpreting how functions behave graphically and numerically.