Question

In Exercises 13-20, use a grapher to (a) identify the domain and range and (b) draw the graph of the function. $$y=\sqrt[3]{x}$$

Short answer

The domain and range of the function \(y=\sqrt[3]{x}\) are both all real numbers (-∞, ∞). The graph of the function is S-shaped and symmetrical about the origin.

Step-by-step solution

Identify the Domain

The domain of the function \(y=\sqrt[3]{x}\) is all real numbers. This is because there is no number that cannot be cube rooted, as negative numbers and zero have cube roots. Therefore, the function is defined for any real number, and hence the domain is (-∞, ∞).

Determine the Range

The range of \(y=\sqrt[3]{x}\) is also all real numbers, because cube rooting any real number will result in a real number. Therefore, the range is (-∞, ∞).

Draw the Graph

In order to draw the graph of the function \(y=\sqrt[3]{x}\), mark a number of points on a graph such that the x-coordinate of each point is a number and the y-coordinate of each point is the cube root of that number. Connect these points in a smooth curve. The graph of \(y=\sqrt[3]{x}\) will look like a slanted S, passing through the origin (0,0), and will rise slower than the graph for a square root function. It is defined for all real numbers and it is symmetric with respect to the origin.