Relations and Functions - Functions

A function $f$ from a set $A$ to a set $B$ is a specific type of relation where every element $a \in A$ has exactly one unique image $b \in B$, denoted as $f(a) = b$. Visually, this is represented by an arrow diagram where every point in the starting set (Domain) has exactly one outgoing arrow, and no point has multiple arrows originating from it.

The domain of a function $f: A \rightarrow B$ is the set $A$, while the codomain is the set $B$. The range is the set of all actual outputs or images, defined as $\{f(x) : x \in A\}$. On a Cartesian coordinate system, the domain is represented by the span of the graph along the $x$-axis, while the range is represented by the span along the $y$-axis.

A Real-Valued Function is a function where the range is a subset of the set of real numbers $\mathbb{R}$. If the domain is also $\mathbb{R}$ or a subset of it, it is called a Real Function. The graph of such a function is a curve or line drawn on the $xy$-plane.

The Identity Function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined by $f(x) = x$ for each $x \in \mathbb{R}$. Its graph is a straight line passing through the origin, bisecting the first and third quadrants at a $45^{\circ}$ angle to both axes.

The Modulus Function $f(x) = |x|$ is defined as $x$ if $x \geq 0$ and $-x$ if $x < 0$. Visually, the graph forms a characteristic 'V' shape with the vertex at the origin $(0,0)$, where the right arm is the line $y = x$ and the left arm is the line $y = -x$.

The Greatest Integer Function $f(x) = [x]$ assumes the value of the greatest integer less than or equal to $x$. Its graph is often called a 'step function' or 'staircase graph' because it consists of horizontal line segments of length 1, where the left endpoint is included (closed circle) and the right endpoint is excluded (open circle).

The Signum Function $f(x)$ returns $1$ if $x > 0$, $0$ if $x = 0$, and $-1$ if $x < 0$. Geometrically, this looks like two disjoint horizontal rays: one at $y=1$ for positive $x$ and one at $y=-1$ for negative $x$, with a single isolated point at the origin $(0,0)$.

Algebra of Functions: For two real functions $f: X \rightarrow \mathbb{R}$ and $g: X \rightarrow \mathbb{R}$, we define addition as $(f+g)(x) = f(x) + g(x)$, subtraction as $(f-g)(x) = f(x) - g(x)$, and multiplication as $(fg)(x) = f(x)g(x)$. For division, $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, provided $g(x) \neq 0$.