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Chapter 1: Problem 107

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

Short Answer

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

Step by step solution

01

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: \(3x + 6 \ge 0\).
02

Solve the inequality

First, isolate the variable on one side of the inequality. Subtract 6 from both sides: \(3x \ge -6\).
03

Divide both sides by 3

To solve for \(x\), divide both sides of the inequality by 3: \(x \ge -2\).
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 \(\textbf{鈭�(expression)}\), 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 \(\textbf{鈭�(3x + 6)}\) 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 \(\textbf{x}\).

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