determine whether the function is bounded above, bounded below, or bounded on its domain\n$y = \\sqrt{7…

determine whether the function is bounded above, bounded below, or bounded on its domain\n$y = \\sqrt{7 - x^{2}}$\nchoose the correct answer below\n○ a. the function is bounded below.\n○ b. the function is bounded above.\n○ c. the function is not bounded.\n○ d. the function is bounded.

determine whether the function is bounded above, bounded below, or bounded on its domain\n$y = \\sqrt{7 - x^{2}}$\nchoose the correct answer below\n○ a. the function is bounded below.\n○ b. the function is bounded above.\n○ c. the function is not bounded.\n○ d. the function is bounded.

Answer

Explanation:

Step1: Analyze the range of the function

For the function (y = \sqrt{7 - x^{2}}), since the square - root function (y=\sqrt{u}) has the property that (y\geq0) (because if (y = \sqrt{u}), then (y^{2}=u\geq0) and (y\geq0)). Also, for the expression inside the square - root (u = 7 - x^{2}), we know that (x^{2}\geq0), so (u=7 - x^{2}\leq7). Then (y=\sqrt{7 - x^{2}}\leq\sqrt{7}).

Step2: Determine the bounds

We have (0\leq y=\sqrt{7 - x^{2}}\leq\sqrt{7}). A function (y = f(x)) is said to be bounded if there exist real numbers (m) and (M) such that (m\leq f(x)\leq M) for all (x) in the domain of (f). Here, (m = 0) and (M=\sqrt{7})

Answer:

D. The function is bounded.