Question

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

Short answer

The domain of the function \(y=\sqrt[3]{x-3}\) is [3, +\infty) and the range is (-\infty, +\infty). The graph is a sideways cubic root curve starting from the point (3,0).

Step-by-step solution

Identify the Domain

The domain of the function is the set of all possible input values, which corresponds to the set of all real numbers in this case. The cubic root function is defined for all real numbers, so the domain is \(D = (-\infty, +\infty)\). However, the function is given as \(y=\sqrt[3]{x-3}\), meaning the value inside the radical (\(x-3\)) can be all real numbers. So for this function, \(x\) must be greater than or equal to 3. So the domain is \(D = [3, +\infty)\)

Identify the Range

The range of the function is the set of all possible output values. Since a cubic root function can provide any real number as an output, the range of this function is \(R = (-\infty, +\infty)\).

Draw the Graph

For drawing the graph of the function, plot various points for values of \(x\) greater than or equal to 3, compute the corresponding \(y=\sqrt[3]{x-3}\), and then plot these points on a graph. The function graph is a sideways cubic root curve starting at the point (3,0).