The modulus, the greatest integer and other special functions

The modulus

The modulus (or absolute value) of $x$ is its distance from zero on the number line. Distance is never negative.

So $|5| = 5$ and $|-5| = 5$. The graph of $y = |x|$ is a "V" with its point at the origin. The domain is $\mathbb{R}$, and the range is $[0, \infty)$.

Some facts that you use all the time:

• $|ab| = |a|\,|b|$, and $\sqrt{x^2} = |x|$. Notice that $\sqrt{x^2}$ is not $x$ when $x$ is negative.
• $|x| \le a$ (with $a > 0$) means $-a \le x \le a$. The distance from zero is at most $a$.
• $|x| \ge a$ means $x \le -a$ or $x \ge a$.
• $|x - c|$ is the distance of $x$ from $c$. So $|x - 2| < 3$ means that $x$ is less than $3$ away from $2$, which is $-1 < x < 5$.

The signum function

The signum function gives only the sign of a number:

The domain is $\mathbb{R}$ and the range is the three-element set $\{-1, 0, 1\}$. Also $x = |x| \operatorname{sgn}(x)$.

The greatest integer function

The greatest integer $[x]$ is the largest integer that is not more than $x$. On the number line, you step to the left until you stand on an integer.

• $[3.7] = 3$, $[5] = 5$, $[0.2] = 0$.
• $[-2.3] = -3$. This one surprises many people. The integers near $-2.3$ are $-3$ and $-2$, and $-3$ is the one on the left, so $[-2.3] = -3$ and not $-2$.

The graph is a staircase. It is flat between two integers, and it jumps by $1$ at every integer. The filled dot shows the value at the integer, and the empty dot shows that the value is not reached. The domain is $\mathbb{R}$ and the range is $\mathbb{Z}$, the integers.

Three facts follow:

• $[x] \le x < [x] + 1$.
• $[x + n] = [x] + n$ for every integer $n$.
• $[x] = x$ exactly when $x$ is an integer.

The fractional part

What is left of $x$ after the whole part is taken away is the fractional part, $\{x\} = x - [x]$. For $x = 3.7$ it is $0.7$. For $x = -2.3$ it is $-2.3 - (-3) = 0.7$. The fractional part is always in $[0, 1)$.

The graph of $\{x\}$ is a row of slanted lines. The pattern repeats after every $1$ unit, so the function is periodic with period $1$.

Slide $x$ across a whole number. The staircase jumps up by $1$ and the sawtooth drops from nearly $1$ back to $0$.

Piecewise functions

All four functions above use different rules on different parts of the domain. A function like this is a piecewise function. For example,

To find $f(-3)$, ask which part holds: $-3 < 0$, so we use $x + 1$ and $f(-3) = -2$. To find $f(0)$ we use the middle part, and $f(0) = 2$. To find $f(3)$ we use $x^2$, and $f(3) = 9$.

When you draw a piecewise function, draw each piece on its own part of the axis. At the ends, use a filled dot where the point belongs to the graph and an empty dot where it does not. A jump in the graph means that the function is not continuous there.

Practise with new numbers

Try it in Python

This program finds the modulus, the signum, the greatest integer and the fractional part of some numbers. Look at the row for $-2.5$.

Remember these

• $|x| = x$ when $x \ge 0$ and $|x| = -x$ when $x < 0$. Also $\sqrt{x^2} = |x|$.
• $|x| < a$ means $-a < x < a$.
• $[x]$ is the greatest integer not more than $x$, so $[-2.3] = -3$.
• $\{x\} = x - [x]$ lies in $[0, 1)$, and it repeats with period $1$.
• A piecewise function uses a different rule on each part of the domain.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find $[4.9]$, $[-4.9]$ and $[-4]$.

2. Solve $|x - 1| \le 4$.

3. Find $\{-1.25\}$.

4. If $f(x) = x + 1$ for $x < 0$ and $f(x) = x^2$ for $x \ge 0$, find $f(-2) + f(2)$.