Question

What is the domain of the variable in the expression \(\sqrt{3 x+6} ?\)

Short answer

The domain is \(x \ge -2\).

Step-by-step solution

Identify the constraint for the square root function

The expression inside the square root, denoted by the radicand, must be non-negative. Therefore, set up the inequality: \(3x + 6 \ge 0\).

Solve the inequality

First, isolate the variable on one side of the inequality. Subtract 6 from both sides: \(3x \ge -6\).

Divide both sides by 3

To solve for \(x\), divide both sides of the inequality by 3: \(x \ge -2\).

Conclusion

The inequality \(x \ge -2\) represents the domain of the variable \(x\) in the given expression. Therefore, \(x\) must be greater than or equal to -2.

Key concepts

square root function

When dealing with square root functions, it's important to remember that the radicand (the expression inside the square root) must be non-negative. This is because the square root of a negative number isn't defined within the set of real numbers. So, whenever you see an expression involving \(\textbf{√(expression)}\), the expression must be greater than or equal to zero to ensure a real number result.

inequality solving

Solving inequalities often involves similar steps to solving equations, but with an additional rule: if you multiply or divide by a negative number, you must flip the inequality sign. In the example \(\textbf{3x + 6 \g\0}\), follow these steps:

• First, isolate the variable by subtracting 6 from both sides to get \(\textbf{3x \g\ -6}\)
• Then, divide the entire inequality by 3 to solve for \(\textbf{x}\). This results in \(\textbf{x \g\ -2}\).

This process helps us determine the range of values that \(\textbf{x}\) can take.

radicand constraint

The constraint on the radicand stems from the fact that the radicand must be non-negative for the square root to be defined in real numbers. In mathematical terms:

• For \(\textbf{√(3x + 6)}\) to be defined, \(\textbf{3x + 6}\) must satisfy \(\textbf{3x + 6 \g\ 0}\).
• This means the real number under the square root cannot be negative, as taking the square root of a negative number results in an imaginary number.

By ensuring that the radicand is non-negative, we ensure the function is properly defined.

isolation of variable

Isolating the variable \(\textbf{x}\) in an inequality or equation is a fundamental step in solving for the domain of a function. Here's a quick breakdown:

• Start with the inequality, \(\textbf{3x + 6 \g\ 0}\).
• Subtract 6 from both sides to get \(\textbf{3x \g\ -6}\).
• Finally, divide both sides by 3 to isolate \(\textbf{x}\), resulting in \(\textbf{x \g\ -2}\).

This technique ensures that we can clearly understand and state the domain of \(\textbf{x}\).