Question

Determine if each value of \(x\) is in the domain of the expression.\(\begin{array}{lll}\sqrt{2 x+4} & \text { (a) } x=-2 & \text { (b) } x=2\end{array}\)

Short answer

Both \(x = -2\) and \(x = 2\) are within the domain of the expression \(\sqrt{2x + 4}\). Therefore, both conditions (a) and (b) are true.

Step-by-step solution

Check for \(x = -2\)

Plug \(x = -2\) into the expression. This gives \(\sqrt{2(-2) + 4} = \sqrt{-4 + 4} = \sqrt{0}\). Since the square root of zero is defined and equals to zero, it means that \(x = -2\) is in the domain of the expression.

Check for \(x = 2\)

Now plug \(x = 2\) into the expression.This gives \(\sqrt{2(2) + 4} = \sqrt{4 + 4} = \sqrt{8}\). The square root of eight is a real number (approximately 2.83), this means value \(x = 2\) is also in the domain of the expression.

Key concepts

Understanding the Square Root Function

The square root function is a fundamental concept in mathematics, often denoted by the symbol \(\sqrt{}\). This function represents the operation of finding a number that, when multiplied by itself, gives the original number inside the square root symbol. For example, since \(3 \times 3 = 9\), we have that \(\sqrt{9} = 3\). However, the square root function is usually concerned only with non-negative values.
One critical aspect of dealing with square root functions is determining which inputs or values are valid. This is known as the domain of the function. For square roots, the domain is limited to values that result in non-negative numbers inside the square root. This is because square roots of negative numbers are not real numbers within the context of basic algebra. Therefore, when you're given an expression such as \(\sqrt{2x + 4}\), you need to ensure that \(2x + 4 \geq 0\) for \(x\) to be part of the domain. This restriction ensures that any \(x\) we plug into the expression results in a valid, real number output.

Exploring Real Numbers

Real numbers include all the numbers that can be found on the number line, encompassing both rational and irrational numbers. This means they include integers, fractions, and decimals that terminate or repeat, as well as numbers that cannot be written as fractions, like \(\pi\) or \(\sqrt{2}\). Real numbers are the basis for many mathematical operations and are key when evaluating expressions.
Importantly, the outputs from operations like square roots need to be real numbers for the result to be considered valid within real-world contexts or basic algebraic scenarios. Negative numbers cannot be the inside value of a square root if you're staying within the realm of real numbers. However, any non-negative result from a square root operation fits within the sea of real numbers.
If you're evaluating whether a number is in the domain of an expression like \(\sqrt{2x + 4}\), you are essentially confirming that the result of \(2x + 4\) is a real, non-negative number, meaning the leave a meaningful conclusion.

Expression Evaluation Insights

Evaluating expressions involves substituting values into an equation or formula and simplifying to reach a result. It's a critical skill in algebra that builds a foundation for understanding functions and their domains. In the case of expressions involving square roots, like \(\sqrt{2x + 4}\), evaluating the expression means checking which \(x\) values will keep the expression real.
Let's consider the expression \(\sqrt{2x + 4}\) for particular values of \(x\). When \(x = -2\), substituting this into the expression gives \(\sqrt{2(-2) + 4} = \sqrt{0}\). Since \(\sqrt{0} = 0\), which is a real number, \(x = -2\) is part of the domain.
Similarly, if you check \(x = 2\), substituting gives \(\sqrt{2(2) + 4} = \sqrt{8}\), which evaluates to a real number approximately equal to 2.83. Hence, \(x = 2\) is also within the domain. This method of evaluation ensures that all outputs of the initial expression are real numbers, confirming valid values for \(x\).

• Substitution of values one at a time helps simplify complex expressions.
• Each evaluation step checks the reality and non-negativity of the result.
• The overall goal is to determine a function's valid inputs by ensuring all conditions are met.