Relations and Functions - Some functions and their graphs: Identity, Constant, Polynomial, Rational, Modulus, Signum, Greatest Integer Function

The Identity Function is defined as $f: \mathbb{R} \to \mathbb{R}$ by $y = f(x) = x$ for every $x \in \mathbb{R}$. The domain and range are both the set of real numbers $\mathbb{R}$. Visually, the graph is a straight line passing through the origin $(0, 0)$, making a $45^\circ$ angle with both the x and y axes, extending infinitely in both directions.

The Constant Function is defined as $f: \mathbb{R} \to \mathbb{R}$ by $y = f(x) = c$, where $c$ is a constant. The domain is $\mathbb{R}$ while the range is the singleton set $\{c\}$. The graph is a horizontal line parallel to the x-axis, crossing the y-axis at the point $(0, c)$. If $c > 0$, the line is above the x-axis; if $c < 0$, it is below.

A Polynomial Function is defined by $f(x) = a_0 + a_1x + a_2x^2 + \dots + a_nx^n$, where $n$ is a non-negative integer and $a_i \in \mathbb{R}$. Common examples include the squaring function $f(x) = x^2$ (a parabola opening upwards with vertex at the origin) and the cubing function $f(x) = x^3$ (an S-shaped curve passing through the origin and extending into the 1st and 3rd quadrants).

A Rational Function is of the form $f(x) = \frac{g(x)}{h(x)}$, where $g(x)$ and $h(x)$ are polynomial functions and $h(x) \neq 0$. For the reciprocal function $f(x) = \frac{1}{x}$, the domain and range are both $\mathbb{R} - \{0\}$. Its graph is a rectangular hyperbola that approaches the axes but never touches them, appearing in the first and third quadrants.

The Modulus Function $f: \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = |x|$, which equals $x$ if $x \ge 0$ and $-x$ if $x < 0$. The domain is $\mathbb{R}$ and the range is the set of non-negative real numbers $[0, \infty)$. The graph is a V-shaped figure with the vertex at the origin, perfectly symmetric about the y-axis.

The Signum Function $f: \mathbb{R} \to \mathbb{R}$ is defined as $f(x) = 1$ if $x > 0$, $f(x) = 0$ if $x = 0$, and $f(x) = -1$ if $x < 0$. The domain is $\mathbb{R}$ and the range is $\{-1, 0, 1\}$. The graph consists of two horizontal half-lines (rays) at $y = 1$ and $y = -1$ starting from the y-axis (excluding the y-axis points) and a single point at the origin $(0, 0)$.

The Greatest Integer Function $f: \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = [x]$, which denotes the greatest integer less than or equal to $x$. For example, $[2.3] = 2$ and $[-2.3] = -3$. The domain is $\mathbb{R}$ and the range is the set of integers $\mathbb{Z}$. The graph looks like a staircase; each step is one unit long, having a closed circle on the left end and an open circle on the right end.