4. Radical Functions & Rational Exponents

Lesson

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.

Positive cube roots greater than $1$1 are smaller than their argument, 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.

The table of values can help us to see this:

$x$x |
$-27$−27 | $-8$−8 | $-1$−1 | $0$0 | $1$1 | $8$8 | $27$27 |
---|---|---|---|---|---|---|---|

$f\left(x\right)=\sqrt[3]{x}$f(x)=^{3}√x |
$-3$−3 | $-2$−2 | $-1$−1 | $0$0 | $1$1 | $2$2 | $3$3 |

Putting these facts together, we should be able to understand why the cubic graph has the shape shown here:

Using the graph we can identify the key characteristics of the square root function, $f\left(x\right)=\sqrt[3]{x}$`f`(`x`)=^{3}√`x`.

Characteristic | $f\left(x\right)=\sqrt[3]{x}$f(x)=^{3}√x |
---|---|

Domain |
Words: $x$ |

Range | Words: $y$y can be any real numberInterval form: $\left(-\infty,\infty\right)$(−∞,∞) |

Extrema | None |

$x$x-intercept |
$\left(0,0\right)$(0,0) |

$y$y-intercept |
$\left(0,0\right)$(0,0) |

Increasing/decreasing | Increasing over its domain |

End behavior |
As $x\to\infty$ As $x\to-\infty$ |

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.

Use this applet below see the different shapes for different powers, as well as how to impose transformations.

What we have already learned about transformations holds true for the cube root function.

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

Consider the graph of the function $y=-\sqrt[3]{x}$`y`=−^{3}√`x`.

Loading Graph...

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

Graph the function $f\left(x\right)=\sqrt[3]{-x}-5$`f`(`x`)=^{3}√−`x`−5.

- Loading Graph...
What is the domain?

All real numbers.

A$x\ge-5$

`x`≥−5B$x\le0$

`x`≤0C$x\ge0$

`x`≥0DWhat is the range?

$y\ge-5$

`y`≥−5AAll real numbers.

B$y\ge0$

`y`≥0C$y\le-5$

`y`≤−5D

Graph functions expressed algebraically and show key features of the graph both by hand and by using technology.