Sets and Functions - Functions, their Domain, Range, and Graphs

A function $f$ from a set $A$ to a set $B$ is a special relation where every element $x \in A$ is associated with exactly one element $y \in B$. Visually, this means in an arrow diagram, every element in the domain has exactly one outgoing arrow. In a coordinate plane, this satisfies the Vertical Line Test: no vertical line intersects the graph of a function at more than one point.

The Domain of a function is the set of all possible input values ($x$) for which the function is defined. For real-valued functions, we must ensure that denominators are non-zero (to avoid division by zero) and expressions under even roots (like $\sqrt{x}$) are non-negative. On a graph, the domain is the horizontal span of the curve along the $x$-axis.

The Range is the set of all actual output values ($f(x)$ or $y$) that the function produces. The Codomain is the set $B$ that contains the range. Visually, the range is represented by the vertical span of the graph along the $y$-axis.

The Identity Function is defined by $f(x) = x$. Its graph is a straight line passing through the origin at a $45^{\circ}$ angle to the $x$-axis, covering all real numbers for both domain and range. The Constant Function $f(x) = c$ results in a horizontal line parallel to the $x$-axis, where the range is the singleton set $\{c\}$.

The Modulus Function $f(x) = |x|$ returns the absolute value of $x$. The graph forms a symmetric 'V' shape with the vertex at the origin $(0,0)$. Its domain is $\mathbb{R}$ and its range is $[0, \infty)$. It is defined piecewise as $x$ if $x \geq 0$ and $-x$ if $x < 0$.

The Signum Function $f(x) = \frac{|x|}{x}$ for $x \neq 0$ and $f(0) = 0$ provides the sign of the input. Its graph consists of two horizontal rays: $y = 1$ for $x > 0$ and $y = -1$ for $x < 0$, along with an isolated point at $(0,0)$. The range is specifically the set $\{-1, 0, 1\}$.

The Greatest Integer Function $f(x) = [x]$ (also known as the floor function) outputs the greatest integer less than or equal to $x$. The graph resembles a series of 'steps' or a 'staircase,' where each step is closed on the left end and open on the right end (e.g., the interval $[1, 2)$ stays at $y=1$).

Algebra of Functions: For two functions $f$ and $g$, we can define $(f+g)(x) = f(x) + g(x)$, $(f-g)(x) = f(x) - g(x)$, and $(f \cdot g)(x) = f(x) \cdot g(x)$. The domain of these new functions is the intersection of the domains of $f$ and $g$, denoted as $D_f \cap D_g$.