Why the Graph of the Modulus Function Is Shaped Like a V

A handful of standard functions — straight lines, parabolas, the modulus, the step-shaped greatest integer function, exponentials and logarithms — appear again and again in algebra, calculus and physics. Knowing their graphs makes domain, range and behaviour visible at a glance.

This part covers functions as rules, polynomial and rational graphs, the modulus, signum and greatest integer functions, and exponential and logarithmic graphs.

**A function is a relation in which each input x has exactly one output y, written $y=f(x)$; its domain is the set of inputs for which the rule is defined, and its range is the set of outputs it actually produces.

Finding the domain. Exclude values that make a denominator zero, place a negative number under a square root, or give the logarithm of a number that is not positive.

Worked example.** Find the domain and range of $f(x)=\sqrt{9-x^2}$.

- Need $9-x^2\ge 0$, so $-3\le x\le 3$; domain $=[-3,3]$
- $9-x^2$ runs from 0 to 9, so $\sqrt{9-x^2}$ runs from 0 to 3; range $=[0,3]$
- For instance, $f(0)=3$ and $f(3)=0$

Second example. For $f(x)=\frac{1}{x-2}$, the domain is $\mathbb{R}-\{2\}$ and the range is $\mathbb{R}-\{0\}$, because the fraction can never equal zero.

An everyday example. An auto-rickshaw fare chart gives exactly one fare for each distance, so fare is a function of distance, with non-negative distances as its domain.

The substance. The domain is read from the rule, but the range must be worked out from the outputs — it is usually the harder of the two.

**A constant function $f(x)=c$ is a horizontal line, the identity function $f(x)=x$ is the line through the origin at $45°$, polynomial functions such as $x^2$ and $x^3$ are smooth unbroken curves, and rational functions such as $\frac{1}{x}$ split into separate branches where the denominator is zero.

Standard graphs:

- Constant** $f(x)=c$ — domain $\mathbb{R}$, range $\{c\}$
- Identity $f(x)=x$ — domain $\mathbb{R}$, range $\mathbb{R}$
- Square $f(x)=x^2$ — an upward parabola; domain $\mathbb{R}$, range $[0,\infty )$
- Cube $f(x)=x^3$ — rises through the origin; domain and range $\mathbb{R}$
- Reciprocal $f(x)=\frac{1}{x}$ — branches in the first and third quadrants; domain and range $\mathbb{R}-\{0\}$

Worked example. For $f(x)=x^2-4x+3$, completing the square gives $f(x)=(x-2{)}^2-1$. The vertex is $(2,-1)$, so the range is $[-1,\infty )$, and the graph cuts the x-axis at $x=1$ and $x=3$.

An everyday example. A flat monthly fee for a phone plan is a constant function of usage, while the area of a square floor tile, $x^2$, is a polynomial function of its side.

The substance. **A polynomial always has domain $\mathbb{R}$, but its range need not be $\mathbb{R}$** — an even-degree polynomial is bounded on one side.

**The modulus function $|x|$ is a V-shape with its corner at the origin, the signum function gives $1$, $0$ or $-1$ according to the sign of x, and the greatest integer function $[x]$ is a staircase that jumps by 1 at every integer.

Modulus function.** $|x|=x$ when $x\ge 0$ and $|x|=-x$ when $x<0$. Domain $\mathbb{R}$, range $[0,\infty )$. The two halves are the lines $y=x$ and $y=-x$, which is why the graph is a V.

Signum function. $\text{sgn}(x)=\frac{|x|}{x}$ for $x\ne 0$, and $\text{sgn}(0)=0$. Domain $\mathbb{R}$, range $\{-1,0,1\}$.

Greatest integer function. $[x]$ is the greatest integer less than or equal to x. Domain $\mathbb{R}$, range $\mathbb{Z}$. Each step includes its left end point and excludes its right one.

Worked example.

- $[3.7]=3$ and $[5]=5$
- $[-2.3]=-3$, because $-3$ is the greatest integer not exceeding $-2.3$
- $\text{sgn}(-4)=-1$ and $|-4|=4$

An everyday example. A courier charge that stays fixed within each kilogram slab and jumps at the next slab follows the staircase pattern of the greatest integer function.

The substance. **$[-2.3]$ is $-3$, not $-2$** — for negative numbers the greatest integer function moves away from zero.

**The exponential function $f(x)=a^x$ with $a>1$ rises steeply, passes through $(0,1)$ and never touches the x-axis, while the logarithmic function $f(x)={\log }_{a}x$ is its mirror image in the line $y=x$, passing through $(1,0)$ and defined only for $x>0$.

Exponential function** $a^x$ with $a>0$, $a\ne 1$:

- Domain $\mathbb{R}$, range $(0,\infty )$
- Passes through $(0,1)$ for every base
- Increasing when $a>1$, decreasing when $0<a<1$

Logarithmic function ${\log }_{a}x$:

- Domain $(0,\infty )$, range $\mathbb{R}$
- Passes through $(1,0)$ for every base
- It is the inverse of $a^x$, so the two graphs are reflections in $y=x$

Worked example. For $f(x)=2^x$: $f(-1)=0.5$, $f(0)=1$ and $f(3)=8$. For $g(x)={\log }_{2}x$: $g(0.5)=-1$, $g(1)=0$ and $g(8)=3$ — the coordinates are swapped. The domain of ${\log }_{10}(x-5)$ is $x>5$.

An everyday example. Money in a bank fixed deposit growing by compound interest follows an exponential curve, and the time needed to double it is found with logarithms.

The substance. **Every exponential graph passes through $(0,1)$ and every logarithmic graph through $(1,0)$** — because $a^0=1$ and ${\log }_{a}1=0$.