Cube Root Functions, Characteristics and Domain and Range
Origin of the cube root
The basic cube root function has the form $y=\sqrt[3]{x}$y=3√x.
Originally the cube root was defined as the side length of a cube whose volume was $x$x, as shown in the diagram.
Thus we have $\sqrt[3]{x}\times\sqrt[3]{x}\times\sqrt[3]{x}=x$3√x×3√x×3√x=x.
Today we define the cube root to include negative numbers, so that for example $\sqrt[3]{-8}=-2$3√−8=−2 and $\sqrt[3]{8}=2$3√8=2.
Describing the graph
Just like square roots, the absolute value of any non-zero cube root of $x$x in the interval $\left(-1,1\right)$(−1,1) is larger than the absolute value of $x$x itself. So for example $\sqrt[3]{0.2}=0.5848...$3√0.2=0.5848... and $\sqrt[3]{-0.2}=-0.5848...$3√−0.2=−0.5848....
Positive cube roots greater than $1$1 are smaller than $x$x. For example $\sqrt[3]{27}=3$3√27=3. Negative cube roots less than $-1$−1 behave similarly, so that $\sqrt[3]{-27}=-3$3√−27=−3.
If we let $f\left(x\right)=\sqrt[3]{x}$f(x)=3√x, then $f\left(-x\right)=-\sqrt[3]{x}$f(−x)=−3√x, and so the function is an odd function. It exhibits rotational symmetry about the origin.
Putting these facts together, and knowing that $\sqrt[3]{0}=0$3√0=0, we should be able to understand why the cubic graph has the shape shown here:
As a comparison, this second graph compares the positions of the graphs of $y=\sqrt{x}$y=√x, $y=\sqrt[3]{x}$y=3√x and $y=\sqrt[4]{x}$y=4√x within the interval $-1\le x\le1$−1≤x≤1. Note that the square root and fourth root functions exist only in the first quadrant.
You can see that in this region $y=\sqrt[4]{x}$y=4√x rises at a faster rate inside this unit square interval than either of $y=\sqrt{x}$y=√x and $y=\sqrt[3]{x}$y=3√x. The graph of $y=\sqrt[3]{x}$y=3√x rises vertically through the origin.
Domain and Range
It is quite clear from the graph that both the domain of $y=\sqrt[3]{x}$y=3√x includes all real $x$x. There are no limits to the range either, since for any value of $y$y, there is a value of $x=\sqrt[3]{y}$x=3√y that can map to it. So the range includes all reals as well.
Worked Examples
question 1
Consider the function $y=\sqrt[3]{x}$y=3√x.
Complete the table of values.
Round any values to two decimal places if necessary.
$x$x $-100$−100 $-10$−10 $-8$−8 $-3$−3 $-1$−1 $0$0 $1$1 $3$3 $8$8 $10$10 $100$100 $y$y $\editable{}$ $-2.15$−2.15 $\editable{}$ $\editable{}$ $-1$−1 $0$0 $\editable{}$ $\editable{}$ $2$2 $2.15$2.15 $4.64$4.64 Which of the following is the graph of $y=\sqrt[3]{x}$y=3√x?
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DIs $y=\sqrt[3]{x}$y=3√x an increasing function or a decreasing function?
Increasing
ADecreasing
B
question 2
Consider the graph of the function $y=-\sqrt[3]{x}$y=−3√x.
Is $y=-\sqrt[3]{x}$y=−3√x an increasing function or a decreasing function?
Increasing function
ADecreasing function
BHow would you describe the rate of decrease of the function?
As $x$x increases, the function decreases at a faster and faster rate.
AAs $x$x increases, the function decreases at a slower and slower rate.
BAs $x$x increases, the function decreases more and more rapidly up to $x=0$x=0, and from $x=0$x=0 onwards, the rate of decrease slows down.
CAs $x$x increases, the function decreases at a constant rate.
D
question 3
Consider the function $y=\sqrt[3]{x-4}$y=3√x−4.
What is the domain of the function?
$x\le3$x≤3
A$x\ge3$x≥3
B$x$x is any real number
CDetermine the range of the function.
$y\le0$y≤0
A$y\ge0$y≥0
B$y$y is any real number
CWhich of the following statements is true?
The function $y=\sqrt[3]{x}$y=3√x increases more rapidly than $y=\sqrt[3]{x-4}$y=3√x−4.
AThe functions $y=\sqrt[3]{x}$y=3√x and $y=\sqrt[3]{x-4}$y=3√x−4 increase at the same rate.
BThe function $y=\sqrt[3]{x-4}$y=3√x−4 increases more rapidly than $y=\sqrt[3]{x}$y=3√x.
C