Question

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

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.