Graphing Cube Root Functions

Problem: Graph \( f(x) = \sqrt[3]{x} \) and find the range of \( f \).

Solution:

Because the domain of \( f \) is the set of all real numbers, we construct a table of values using convenient inputs whose cube roots are integers:

The values of \( x \) were selected so that their cube roots are whole numbers, making it easy to plot the points.

• Range: \( (-\infty, +\infty) \)

Problem: Graph \( f(x) = \sqrt[3]{x - 2} \) and find the range of \( f \).

Solution:

The domain of the cube root function is the set of all real numbers. It is easy to calculate \( \sqrt[3]{x - 2} \) if you select values of the argument \( x - 2 \) equal to \(-8, -1, 0, 1, 8\), and then solve for \( x \):

The last two rows in the table of data are used to graph \( f \).

• Range: \( (-\infty, +\infty) \)

Note: The graph of \( f(x) = \sqrt[3]{x - 2} \) is that of \( f(x) = \sqrt[3]{x} \) shifted \( 2 \) units to the right.

Problem: Graph \( f(x) = -\sqrt[3]{x + 1} \) and find the range of \( f \).

Solution:

The domain of the function is the set of all real numbers. We select values of \( x + 1 \) equal to \(-8, -1, 0, 1, 8\) to construct the table of values and solve for \( x \):

• Range: \( (-\infty, +\infty) \)

Problem: Graph \( f(x) = -2\sqrt[3]{x - 2} + 2 \) and find the range of \( f \).

Solution:

The domain of function \( f \) is the set of all real numbers. We select values of \( x - 2 \) equal to \(-8, -1, 0, 1, 8\) to construct the table:

• Range: \( (-\infty, +\infty) \)