Graphing Cube Root Functions

A tutorial on graphing and sketching cube root functions is presented. The graph, domain, range of these functions and other properties are discussed.

Graph, Domain and Range of the Basic Cube Root Function: f(x) = ∛x

The domain of function f defined by f(x) = ∛x is the set of all real numbers.
The range of f is the set of all real numbers.

Example 1

Graph f( x ) = ∛x and find the range of f.
Solution to Example 1
Because the domain of f is the set of all real numbers, we might construct a table of values as follows:

x	-8	1	0	1	8
f(x) = ∛x	-2	1	0	1	2

The values of x were selected so that the cube root of these values are whole numbers which make it easy to plot the points shown in the table.
points and graph of cube root(x)

The range of f is given by the interval (-? , +?).

Graph the More General Cube Root Function: f(x) = ∛x

Example 2

Graph
f( x ) = ∛ (x - 2)
and find the range of f.
Solution to Example 2
The domain of the cube root function given above is the set of all real numbers.
It easy to calculate ∛ (x - 2)if you select values of (x - 2) as -8, -1, 0, 1 and 8 to construct a table of values then find x in order to graph f.

x - 2	-8	-1	0	1	8
f(x) = ∛ (x - 2)	-2	-1	0	1	2
x	-6	1	2	3	10

The last two rows in the table of data are used to graph f.
points and graph of cube root(x - 2)

The range of f is the set of all real numbers.
Note also that the graph of f(x) = ∛ (x - 2) is that of f(x) = ∛ ( x ) shifted 2 units to the right.

Example 3

Graph f( x ) = - ∛(x + 1) and find the range of f.
Solution to Example 3
The domain of the function given above is the set of all real numbers
We now select values of (x + 1) as -8, -1, 0, 1 and 8 to construct a table of values then find x in order to graph f .

x + 1	-8	-1	0	1	8
f(x) = - ∛ (x + 1)	2	1	0	- 1	- 2
x	-9	-2	-1	0	7

The range of f is given by the interval (-? , +?).

Example 4

Graph f( x ) = - 2 ∛ (x - 2) + 2 and find the range of f.

Solution to Example 4
The domain of function f is the set of all real values.
We now select values of (x - 2) as -8, -1, 0, 1 and 8 to construct a table of values.

x - 2	-8	-1	0	1	8
f(x) = - 2 ∛ (x - 2) + 2	6	4	2	0	- 2
x	-6	1	2	3	10

points and graph of - 2 cube root (x - 2) + 2

The range of f is the set of all real numbers.