Question

In Exercises 13-20, use a grapher to (a) identify the domain and range and (b) draw the graph of the function. $$y=\sqrt{9-x^{2}}$$

Short answer

The domain of the function is \([-3, 3]\), the range is \([0, 3]\), and the graph of the function resembles the upper half of a circle of radius 3, centered at the origin.

Step-by-step solution

Identify the Domain

The domain of a function is the set of all x-values that can possibly go into the function. Looking at our function \(y=\sqrt{9-x^{2}}\), x-values can be anything that makes \(9-x^{2}\) non-negative (since we can't take the square root of a negative number). Hence, we can write an inequality \(9-x^{2} \geq 0\). Solving it we get x-values in the range \([-3, 3]\). Therefore, the domain is \([-3, 3]\).

Identify the Range

The range of a function are the possible y-values that can come out of the function. Since our function is a square root function, and we know we can't get negative values out of a square root function, the smallest possible value for y is 0. Since we're taking the square root of \(9-x^{2}\), y can be as large as \(sqrt{9} = 3\). Therefore, our range is from 0 to 3, inclusive, or [0, 3].

Draw the Graph

We use a graphing tool to graph the function. By setting up a table of x-values in the domain and correlating y-values obtained by inserting each x-value into the equation, we outline the graph. The graph will resemble the upper half of a circle (due to the nature of the original function) with radius 3, centered at the origin.