Question

what is the domain of the function $y = 2\sqrt{x - 6}$?
options:
$-\infty < x < \infty$
$0 \leq x < \infty$
$3 \leq x < \infty$
$6 \leq x < \infty$

Explanation:

Step1: Recall domain of square root

For a square root function $\sqrt{f(x)}$, the expression inside the square root, $f(x)$, must be non - negative (i.e., $f(x)\geq0$) because the square root of a negative number is not a real number. In the function $y = 2\sqrt{x - 6}$, the expression inside the square root is $x-6$.

Step2: Solve the inequality

We set up the inequality $x - 6\geq0$. To solve for $x$, we add 6 to both sides of the inequality. Using the addition property of inequalities, if we have $x-6\geq0$, then $x-6 + 6\geq0 + 6$, which simplifies to $x\geq6$. In interval notation, this is $6\leq x<\infty$.

Answer:

$6\leq x<\infty$ (the option with the inequality $6\leq x<\infty$)