Cubic and Other Integer-Power Functions, foundations for calculus lesson

A cube with side $2$ cm holds $8$ cubic centimeters, and one with side $4$ cm holds $64$ . Doubling the side multiplies the volume by eight, and that rapid growth is the whole personality of a power function. Small exponents and large ones, even ones and odd ones, all follow a few simple rules about sign, symmetry and size that you can read straight off the formula.

Key terms

Integer-power function

Definition An integer-power function, in this lesson, is a function of the form

f(x) = x^n,

where $n$ is a fixed positive integer. Here $x^n$ means $x$ multiplied by itself $n$ times, so it is defined for every real $x$ , including negative $x$ and $x = 0$ . The exponent $0$ and negative exponents are not part of this family. The function $x^0$ is the constant $1$ away from $0$ , and $x^{-1} = \frac{1}{x}$ is the reciprocal function, which has its own lesson.

In plain words Pick a whole number $n$ and raise the input to that power. The cases $n = 1$ , $2$ and $3$ give the line $y = x$ , the parabola $y = x^2$ and the cubic $y = x^3$ .

Example For $f(x) = x^5$ the values at $x = -2$ , $-1$ , $0$ , $1$ and $2$ are $-32$ , $-1$ , $0$ , $1$ and $32$ . The sign of the output matches the sign of the input, and the outputs grow fast once $\lvert x \rvert > 1$ .

Cubic function

Definition A cubic function is a polynomial function of degree $3$ ,

\begin{gathered} f(x) = ax^3 + bx^2 + cx + d, \\[0.4em] a \ne 0, \end{gathered}

with real coefficients $a$ , $b$ , $c$ and $d$ . Its domain is all real numbers. The simplest one, $f(x) = x^3$ , is the parent cubic.

In plain words A cubic is any formula whose highest power of $x$ is $x^3$ . Far to the right and far to the left its outputs go in opposite directions. For large $\lvert x \rvert$ the term $ax^3$ outweighs the others, and $x^3$ has the sign of $x$ .

Example To sketch $f(x) = x^3 - 4x$ , factor first.

x^3 - 4x = x(x - 2)(x + 2).

The zeros are $-2$ , $0$ and $2$ . Counting negative factors tells you the sign between them, and a few values confirm it.

$x$	$-3$	$-1$	$1$	$3$
$f(x)$	$-15$	$3$	$-3$	$15$

So the graph comes up from below, crosses at $-2$ , rises above the axis, comes back down through the origin, dips below the axis, and climbs back up through the axis at $2$ . A table cannot show exactly where the turns are. Finding those needs derivatives later.

Even power

Definition If $n$ is an even positive integer, then for every real $x$

(-x)^n = x^n \qquad \text{and} \qquad x^n \ge 0,

with $x^n = 0$ only at $x = 0$ . The reason is that an even number of negative factors multiply to a positive number. For even $n$ and $\lvert x \rvert < 1$ with $x \ne 0$ , higher even powers are smaller, and for $\lvert x \rvert > 1$ they are larger.

In plain words An even power wipes out the sign of the input. The graphs of $x^2$ , $x^4$ and $x^6$ are U shapes with their lowest point at the origin. The higher the power, the flatter the bottom between $-1$ and $1$ and the steeper the sides beyond them.

Example Compare $x^2$ and $x^4$ . Their difference factors as

x^4 - x^2 = x^2(x - 1)(x + 1).

For $0 < \lvert x \rvert < 1$ this is negative, so $x^4 < x^2$ . For $\lvert x \rvert > 1$ it is positive, so $x^4 > x^2$ . The graphs meet only at $x = -1$ , $0$ and $1$ .

$x$	$\frac{1}{2}$	$1$	$2$
$x^2$	$\frac{1}{4}$	$1$	$4$
$x^4$	$\frac{1}{16}$	$1$	$16$

The same values appear at $-\frac{1}{2}$ , $-1$ and $-2$ . Near the origin $x^4$ hugs the axis more closely, and past $1$ it shoots up faster.

Odd power

Definition If $n$ is an odd positive integer, then for every real $x$

(-x)^n = -x^n,

so $x^n$ has the same sign as $x$ . Moreover, for every real number $y$ there is exactly one real $x$ with $x^n = y$ , called the real $n$ th root of $y$ .

In plain words An odd power keeps the sign of the input. The graphs of $x^3$ , $x^5$ and $x^7$ run from the lower left, through the origin, to the upper right.

Example The equation $x^3 = -27$ has the single real solution $x = -3$ , since $(-3)^3 = -27$ and no other real number cubes to $-27$ . By contrast, the even power equation $x^4 = 16$ has two real solutions, $x = 2$ and $x = -2$ , and $x^4 = -16$ has none.

Symmetry

Definition A function $f$ whose domain contains $-x$ whenever it contains $x$ is called even if $f(-x) = f(x)$ for every $x$ in its domain, and odd if $f(-x) = -f(x)$ for every $x$ in its domain. The graph of an even function is symmetric about the $y$ axis. The graph of an odd function is symmetric about the origin, meaning $(x, y)$ is on the graph exactly when $(-x, -y)$ is. Most functions are neither even nor odd.

In plain words An even graph is its own mirror image across the $y$ axis. An odd graph looks the same after a half turn about the origin. So $x^n$ is even when $n$ is even and odd when $n$ is odd, which is where the names come from.

Example The function $f(x) = x^3 - 4x$ from above is odd, because

\begin{aligned} f(-x) &= (-x)^3 - 4(-x) \\[0.4em] &= -x^3 + 4x = -f(x). \end{aligned}

That matches the table, where $f(-3) = -15$ and $f(3) = 15$ . The function $g(x) = x^3 + 1$ is neither, since $g(1) = 2$ while $g(-1) = 0$ , which equals neither $2$ nor $-2$ .

Increasing function

Definition A function $f$ is increasing on an interval $I$ if $f(a) < f(b)$ whenever $a$ and $b$ are in $I$ with $a < b$ . It is decreasing on $I$ if $f(a) > f(b)$ whenever $a < b$ in $I$ . The function $x^3$ is increasing on all of $\mathbb{R}$ . To see it, take $a < b$ and factor.

b^3 - a^3 = (b - a)(a^2 + ab + b^2).

The first factor is positive. Completing the square gives

a^2 + ab + b^2 = \left(a + \frac{b}{2}\right)^2 + \frac{3b^2}{4},

which is positive unless $a = b = 0$ , and that is impossible when $a < b$ . So $b^3 > a^3$ . The same holds for every odd power. An even power is decreasing on $(-\infty, 0]$ and increasing on $[0, \infty)$ .

In plain words Increasing means the graph always goes up as you move right. The cube curve never turns back, even though it flattens out near the origin.

Example Take $a = -2$ and $b = 1$ . Then $a^3 = -8 < 1 = b^3$ , as the rule says. For $x^2$ the same inputs give $4$ and $1$ , so the order flips, which shows that $x^2$ is not increasing on any interval that contains both negative and positive numbers.

Domain and range

Definition For every positive integer $n$ the domain of $f(x) = x^n$ is $\mathbb{R}$ . If $n$ is odd, the range is $\mathbb{R}$ , because each real $y$ is the output at its real $n$ th root. If $n$ is even, the range is $[0, \infty)$ , because every output is nonnegative and each $y \ge 0$ is the output at $x = \sqrt[n]{y}$ , the nonnegative $n$ th root.

In plain words Every real number can go into a power function. An odd power can produce any real number, while an even power produces exactly the numbers that are zero or positive.

Example The function $h(x) = x^4 - 3$ has domain $\mathbb{R}$ . Since $x^4$ takes every value in $[0, \infty)$ , the outputs of $h$ fill $[-3, \infty)$ , with the smallest value $-3$ at $x = 0$ . The output $13$ is reached at $x = 2$ and $x = -2$ .

Check yourself

Decide whether $f(x) = x^5 - x$ is even, odd or neither, and find $f(2)$ and $f(-2)$ .

Show the answer

It is odd, because $f(-x) = -x^5 + x = -f(x)$ . The values are $f(2) = 32 - 2 = 30$ and $f(-2) = -30$ .

Which is larger, $\left(\frac{1}{2}\right)^3$ or $\left(\frac{1}{2}\right)^5$ ? Which is larger, $3^3$ or $3^5$ ?

Show the answer

For an input between $0$ and $1$ more factors make the product smaller, so $\left(\frac{1}{2}\right)^3 = \frac{1}{8}$ is larger than $\left(\frac{1}{2}\right)^5 = \frac{1}{32}$ . Above $1$ the order reverses, and $3^5 = 243$ is larger than $3^3 = 27$ .

Find the range of $g(x) = (x - 1)^3 + 2$ and the input that gives the output $10$ .

Show the answer

The cube $(x - 1)^3$ takes every real value, so the range is $\mathbb{R}$ . Setting $(x - 1)^3 + 2 = 10$ gives $(x - 1)^3 = 8$ , so $x - 1 = 2$ and $x = 3$ . It is the only solution because cubing is increasing.

Cubic and Other Integer-Power Functions

Key terms

Integer-power function

Cubic function

Even power

Odd power

Symmetry

Increasing function

Domain and range

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus