What is the domain of the function $y=2 \sqrt{x-6}$ ?
A. $-\infty < x < \infty$
B. $0 \leq x<\infty$
C. $3 \leq x<\infty$
D. $6 \leq x<\infty$
Answer (2)
The domain of y = 2 x − 6 requires the expression inside the square root to be non-negative.
Set up the inequality x − 6 ≥ 0 .
Solve the inequality to find x ≥ 6 .
Express the solution in interval notation: 6 ≤ x < ∞ .
Explanation
Understanding the Problem We want to find the domain of the function y = 2 x − 6 . The domain is the set of all possible x values for which the function is defined. Since we have a square root, the expression inside the square root must be non-negative.
Setting up the Inequality The expression inside the square root must be greater than or equal to zero: x − 6 ≥ 0
Solving the Inequality To solve the inequality x − 6 ≥ 0 , we add 6 to both sides: x − 6 + 6 ≥ 0 + 6 x ≥ 6
Expressing the Solution The domain of the function is all x values such that x ≥ 6 . In interval notation, this is written as [ 6 , ∞ ) . Since the question asks for the domain in the form a ≤ x < ∞ , the correct answer is 6 ≤ x < ∞ .
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if x represents the number of hours a machine operates, and the function y represents the output of the machine, then the domain tells us the valid range of operating hours. If the function is y = 2 x − 6 , it means the machine must operate for at least 6 hours before it starts producing any output. This concept is also applicable in finance, where x could represent the investment amount, and the function gives the return on investment, with a minimum investment required before any returns are realized.
The domain of the function y = 2 x − 6 is determined by the condition that the expression inside the square root must be non-negative. Therefore, solving the inequality gives us x ≥ 6 . The chosen answer is option D: 6 ≤ x < ∞ .
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