The Cube-Root Function, foundations for calculus lesson

A box shaped like a cube holds $27$ liters, and you want its edge length. That is the cube root, $\sqrt[3]{27} = 3$ . Unlike the square root, the cube root has no trouble with negative numbers, because a negative number cubed is negative. So its graph has no endpoint. It runs from the lower left, through the origin, to the upper right.

Key terms

Real cube root

Definition For every real number $a$ , the real cube root $\sqrt[3]{a}$ is the unique real number $b$ with

b^3 = a.

Such a $b$ exists because $x^3$ takes every real value, and it is unique because $x^3$ is increasing on $\mathbb{R}$ , so two different numbers never have the same cube. It follows that

\left(\sqrt[3]{x}\right)^3 = x \qquad \text{and} \qquad \sqrt[3]{y^3} = y

for every real $x$ and every real $y$ , negative values included.

In plain words The cube root of a number is the one real number that cubes to it. There is always exactly one, so no choice of sign is needed.

Example Since $4^3 = 64$ , the cube root of $64$ is $4$ . Since $\left(\frac{2}{5}\right)^3 = \frac{8}{125}$ , the cube root of $\frac{8}{125}$ is $\frac{2}{5}$ . Compare with square roots, where $\sqrt{(-5)^2} = 5$ . For cube roots, $\sqrt[3]{(-5)^3} = \sqrt[3]{-125} = -5$ , and the sign survives.

Negative input

Definition For every real $a$ ,

\sqrt[3]{-a} = -\sqrt[3]{a},

because $\left(-\sqrt[3]{a}\right)^3 = -a$ and the real cube root is unique. In particular the cube root of a negative number is a negative real number. Be careful with calculators and software. Many of them compute a fractional power such as $(-8)^{1/3}$ with a complex principal value, about $1 + 1.732i$ , instead of the real root $-2$ , or report an error. When graphing, use a dedicated real cube root, or compute $-\sqrt[3]{8}$ by hand.

In plain words Take the cube root of the size of the number and put the minus sign back. Negative inputs are perfectly fine.

Example The cube root of $-27$ is $-3$ , since $(-3)^3 = -27$ . The cube root of $-0.001$ is $-0.1$ , since $(-0.1)^3 = -0.001$ .

Cube-root function

Definition The cube-root function is $f(x) = \sqrt[3]{x}$ , where $\sqrt[3]{x}$ is the real cube root. It is the inverse function of $g(x) = x^3$ on $\mathbb{R}$ , so its graph is the reflection of the cube curve $y = x^3$ across the line $y = x$ .

In plain words Cubing and taking the cube root undo each other completely. Swap the coordinates of every point on $y = x^3$ and you get the cube-root curve.

Example The points $(-2, -8)$ , $(1, 1)$ and $(2, 8)$ on $y = x^3$ become $(-8, -2)$ , $(1, 1)$ and $(8, 2)$ on $y = \sqrt[3]{x}$ . Where the cube curve is steep, the root curve is flat, and the reverse.

Domain and range

Definition The domain of $f(x) = \sqrt[3]{x}$ is $\mathbb{R}$ and its range is $\mathbb{R}$ . Every real input has a real cube root, and every real $y$ is an output, at the input $x = y^3$ , since $\sqrt[3]{y^3} = y$ .

In plain words No input is forbidden and no output is out of reach. The graph extends forever to the left, right, up and down.

Example To get the output $-1.5$ , use the input $(-1.5)^3 = -3.375$ . Check that $\sqrt[3]{-3.375} = -1.5$ , since $1.5^3 = 3.375$ .

Increasing

Definition A function $f$ is increasing on an interval $I$ if $f(a) < f(b)$ whenever $a$ and $b$ are in $I$ with $a < b$ . The cube-root function is increasing on $\mathbb{R}$ . Suppose $a < b$ . If $\sqrt[3]{a} \ge \sqrt[3]{b}$ held, then cubing, which preserves order on all of $\mathbb{R}$ , would give $a \ge b$ , a contradiction. So

a < b \quad \text{implies} \quad \sqrt[3]{a} < \sqrt[3]{b}.

In plain words Bigger input, bigger cube root, with no exceptions, even across zero. The graph rises all the way from left to right.

Example The inputs $-8 < -1 < 0 < 1 < 8$ give cube roots $-2 < -1 < 0 < 1 < 2$ , in the same order. Near the origin the curve is very steep. The tiny input $0.001$ already has cube root $0.1$ , a hundred times larger. Farther out it flattens, and it takes the input $8$ just to reach the output $2$ .

Origin symmetry

Definition A graph has origin symmetry when $(x, y)$ is on it exactly when $(-x, -y)$ is. For a function with domain symmetric about $0$ this means $f(-x) = -f(x)$ for every $x$ , that is, $f$ is odd. The cube-root function is odd, by the rule $\sqrt[3]{-x} = -\sqrt[3]{x}$ .

In plain words Turn the graph half a turn around the origin and it lands on itself. The left half is the right half flipped both ways.

Example The point $(27, 3)$ is on the graph, so $(-27, -3)$ is too. Plotting the right half carefully gives you the left half for free.

Coordinate pair

Definition A coordinate pair $(a, b)$ lies on the graph of $f$ exactly when $f(a) = b$ . For a formula such as $g(x) = \sqrt[3]{x - 2} + 1$ , exact pairs are easiest to find by choosing the inside $x - 2$ to be a perfect cube, $-8$ , $-1$ , $0$ , $1$ or $8$ , and then solving for $x$ . More generally, every real $y$ is an output of $g$ , at the input

x = 2 + (y - 1)^3.

In plain words Decide what the cube root should come out to, then work backwards to the input. Each choice gives a point you can plot exactly.

Example For $g(x) = \sqrt[3]{x - 2} + 1$ the inner cubes give this table.

Inside $x - 2$	$-8$	$-1$	$0$	$1$	$8$
Input $x$	$-6$	$1$	$2$	$3$	$10$
Output $g(x)$	$-1$	$0$	$1$	$2$	$3$

The graph is the cube-root curve moved $2$ right and $1$ up. Its center of symmetry moves from the origin to $(2, 1)$ , and the domain and range stay $\mathbb{R}$ . The points locate the curve, and the fact that $g$ is increasing tells you how to join them.

Check yourself

Find the cube root of $-\frac{27}{64}$ .

Show the answer

The cube of $-\frac{3}{4}$ is $-\frac{27}{64}$ , so the cube root is $-\frac{3}{4}$ , or $-0.75$ .

Solve $\sqrt[3]{x - 2} + 1 = -2$ .

Show the answer

Subtracting $1$ gives $\sqrt[3]{x - 2} = -3$ . Cubing both sides gives $x - 2 = -27$ , so $x = -25$ . Cubing cannot add false solutions because it is one-to-one, and indeed $\sqrt[3]{-27} + 1 = -2$ .

For $y = -5$ , compare $\sqrt[3]{y^3}$ with $\sqrt{y^2}$ .

Show the answer

The cube root gives $\sqrt[3]{-125} = -5$ , the original number. The square root gives $\sqrt{25} = 5$ , which is $\lvert -5 \rvert$ . Cube roots keep the sign and square roots do not.

The Cube-Root Function

Key terms

Real cube root

Negative input

Cube-root function

Domain and range

Increasing

Origin symmetry

Coordinate pair

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus