Question

Does the domain of \(y=\sqrt{x+3}\) include negative values of \(x ?\) Explain.

Short answer

Yes, the domain of the function \(y=\sqrt{x+3}\) does include negative values of \(x\), but only those that are greater than or equal to -3.

Step-by-step solution

Identify Given

The given function is \(y=\sqrt{x+3}\)

Analyze the Domain in General

The domain of a function consists of all the input or the set of allowable \(x\)-values. For the function \(y=\sqrt{x+a}\), where \(a\) is any real number, we cannot have negative numbers under the square root in the real number system.

Analyze the Domain of Given Function

For function \(y=\sqrt{x+3}\), we should solve \(x+3 \geq 0\) for \(x\). In doing so, we find that \(x \geq -3\). Therefore, negative values of \(x\) that are greater than or equal to -3 are included in the domain.

Key concepts

Square Root Function

The square root function is a fundamental mathematical concept represented as \( y = \sqrt{x} \), where \( x \) is a variable representing any non-negative real number. This function takes in a value and produces its square root, which is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, since \( 3 \times 3 = 9 \).

Understanding the nature of square root functions is essential because they often appear in various branches of mathematics, physics, and engineering. In the context of the given exercise, the function \( y = \sqrt{x + 3} \) is a modified version of the standard square root function. Here, the function shifts the input of the square root by adding 3 to the variable \( x \) before taking the square root. The implications of this shift are important when considering the domain of the function, which leads us into domain analysis.

Domain Analysis

Performing a domain analysis on a function involves determining the set of all possible input values (\( x \)-values) that the function can accept without causing issues, such as undefinedness or complex numbers when dealing with real functions. Particularly for square root functions like \( y = \sqrt{x + 3} \), the input values under the square root sign must be non-negative since the real square root of a negative number is not a real number.

In our exercise, to analyze the domain of \( y = \sqrt{x + 3} \), we must ensure that the expression inside the square root, \( x + 3 \), is greater than or equal to zero, or \( x + 3 \geq 0 \). Solving for \( x \) gives us \( x \geq -3 \), meaning the domain includes all real numbers greater than or equal to -3. This includes negative values of \( x \), as long as they are not less than -3. Therefore, the domain is the interval \( [-3, \infty) \), where the square bracket indicates that -3 is included in the domain.

Real Number System

The real number system is a continuous number system that includes all the numbers on the number line, encompassing both positive and negative numbers, fractions, and irrational numbers such as \( \pi \) and \( \sqrt{2} \). They are called 'real' because they are not imaginary, which means every point on the number line corresponds to a real number, and every real number corresponds to a point.

In the context of the exercise, the real number system dictates that the domain of square root functions can only include non-negative numbers under the square root to remain within the system. This is why the function \( y = \sqrt{x + 3} \) has a domain restricted to values of \( x \) that satisfy \( x + 3 \geq 0 \), ensuring that all possible values for \( x \) result in a real number output for \( y \). This fundamental property ensures that the outputs are meaningful within the context of the real-world scenarios where such mathematical functions are applied.