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`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ The graph of $y=\sqrt[3]{x}$

`y`=^{3}√`x`is given.Loading Graph...Is $y=\sqrt[3]{x}$

`y`=^{3}√`x`an increasing function or a decreasing function?Increasing

ADecreasing

BIncreasing

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

BIncreasing 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.DAs $x$

`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`≥0DAll 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$y\ge-5$

`y`≥−5AAll real numbers.

B$y\ge0$

`y`≥0C$y\le-5$

`y`≤−5D

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. '''Include rational, square root and cube root; emphasize selection of appropriate models.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. '''Include rational, square root and cube root; emphasize selection of appropriate models.

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. '''Focus on using key features to guide selection of appropriate type of model function

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. '''Focus on using key features to guide selection of appropriate type of model function