Foundations of Functions, Graphs, Symmetries & Inverses

1. The Real Number Continuum $\mathbb{R}$ & Set Notation

The foundation of all single-variable real analysis and calculus is the set of real numbers $\mathbb{R}$, equipped with the standard algebraic operations of addition and multiplication, satisfying the field axioms, order axioms, and the Completeness Axiom (every non-empty subset of $\mathbb{R}$ bounded above has a least upper bound or supremum in $\mathbb{R}$).

Subsets of $\mathbb{R}$ are frequently represented using interval notation:

2. Formal Definition of a Real Function

The Range (or image) of $f$ is the set of all attained values:

3. The Natural Domain & The Vertical Line Test

Unless explicitly restricted, the natural domain of a function defined by an algebraic expression is the largest subset of $\mathbb{R}$ for which the expression produces a well-defined real number. In single-variable calculus, two foundational constraints govern natural domains:

1. Division by Zero: Denominators cannot equal zero ($Q(x) \ne 0$).
2. Even Roots of Negative Numbers: For $\sqrt[2n]{g(x)}$, we require $g(x) \ge 0$.

The Vertical Line Test: A curve in the Cartesian plane $\mathbb{R}^2$ represents the graph of a function $y = f(x)$ if and only if no vertical line $x = c$ intersects the curve at more than one point. If a vertical line intersects at two or more points, a single input would map to multiple outputs, violating single-valued functionality.

1. Polynomial Functions

A polynomial function of degree $n \in \mathbb{N}_0$ with real coefficients $a_n, a_{n-1}, \dots, a_0$ ($a_n \ne 0$) is defined for all $x \in \mathbb{R}$ by:

By the Fundamental Theorem of Algebra, a degree-$n$ polynomial has exactly $n$ complex roots (counting multiplicity), and at most $n$ distinct real roots. The end behavior of $P(x)$ as $x \to \pm\infty$ is strictly governed by its leading monomial term $a_n x^n$:

2. Rational Functions & Asymptotes

A rational function is a ratio of two polynomials $R(x) = \frac{P(x)}{Q(x)}$ where $Q(x) \not\equiv 0$. Its natural domain is $\operatorname{Dom}(R) = \{ x \in \mathbb{R} : Q(x) \ne 0 \}$.

Let $P(x) = a_n x^n + \dots$ and $Q(x) = b_m x^m + \dots$ with $a_n \ne 0, b_m \ne 0$:

• Vertical Asymptotes: If $x = c$ is a root of $Q(x)$ such that $(x - c)$ has higher multiplicity in $Q(x)$ than in $P(x)$, then $\lim_{x \to c} |R(x)| = \infty$, producing a vertical asymptote at $x = c$. If $(x - c)$ cancels completely, $x = c$ is a removable discontinuity (hole).
• Horizontal Asymptotes:
  • If $n < m$: The line $y = 0$ is a horizontal asymptote ($\lim_{x \to \pm\infty} R(x) = 0$).
  • If $n = m$: The horizontal line $y = \frac{a_n}{b_m}$ is an asymptote ($\lim_{x \to \pm\infty} R(x) = a_n / b_m$).
  • If $n > m$: No horizontal asymptote exists.
• Oblique (Slant) Asymptotes: If $n = m + 1$, polynomial long division yields $R(x) = (m_0 x + c_0) + \frac{r(x)}{Q(x)}$ with $\deg(r) < m$. The line $y = m_0 x + c_0$ is an oblique asymptote as $x \to \pm\infty$.

1. Piecewise-Defined Functions

A function defined by different analytical rules on disjoint sub-intervals of its domain is called a piecewise-defined function:

2. The Absolute Value (Modulus) Function

The absolute value function $|x|: \mathbb{R} \to [0, \infty)$ is defined algebraically as:

Fundamental Properties of Absolute Value:

1. Non-negativity: $|x| \ge 0$, with $|x| = 0 \iff x = 0$.
2. Multiplicativity: $|x y| = |x| |y|$ and $\left|\frac{x}{y}\right| = \frac{|x|}{|y|}$ for $y \ne 0$.
3. The Triangle Inequality: For all $x, y \in \mathbb{R}$,
   $$|x + y| \le |x| + |y|$$
4. The Reverse Triangle Inequality:
   $$||x| - |y|| \le |x - y|$$
5. Interval Equivalence: For any $c > 0$, $|x - a| < c \iff a - c < x < a + c$.

3. Step, Signum & Floor Functions

The signum function $\operatorname{sgn}(x)$ extracts the algebraic sign of $x$:

The floor function (greatest integer function) $\lfloor x \rfloor$ maps $x$ to the unique integer $k \in \mathbb{Z}$ satisfying $k \le x < k + 1$. The ceiling function $\lceil x \rceil$ satisfies $k - 1 < x \le k$.

1. Rigid and Non-Rigid Transformations

Given the base graph $y = f(x)$, algebraic modifications to the argument or output produce exact geometric transformations in $\mathbb{R}^2$ ($c > 0$):

• Vertical Shift: $y = f(x) + c$ shifts the graph upward by $c$ units; $y = f(x) - c$ shifts downward.
• Horizontal Shift: $y = f(x - c)$ shifts the graph to the right by $c$ units; $y = f(x + c)$ shifts to the left.
• Vertical Scaling & Reflection: $y = a f(x)$ stretches vertically by factor $|a|$ if $|a| > 1$, compresses if $0 < |a| < 1$, and reflects across the $x$-axis if $a < 0$.
• Horizontal Scaling & Reflection: $y = f(b x)$ compresses horizontally by factor $1/|b|$ if $|b| > 1$, stretches if $0 < |b| < 1$, and reflects across the $y$-axis if $b < 0$.

2. Algebraic Symmetry Tests: Even and Odd Functions

Decomposition Theorem: Every function $f: \mathbb{R} \to \mathbb{R}$ whose domain is symmetric about the origin can be uniquely decomposed into the sum of an even function $f_{\text{even}}$ and an odd function $f_{\text{odd}}$:

3. Monotonicity on Intervals

Let $I \subseteq \operatorname{Dom}(f)$ be an interval. We define:

• Strictly Increasing: $\forall x_1, x_2 \in I$, $x_1 < x_2 \implies f(x_1) < f(x_2)$.
• Strictly Decreasing: $\forall x_1, x_2 \in I$, $x_1 < x_2 \implies f(x_1) > f(x_2)$.
• Strictly Monotonic: A function that is either strictly increasing on $I$ or strictly decreasing on $I$. Strictly monotonic functions are guaranteed to be one-to-one (injective) on $I$.

1. Composition of Functions

Given two functions $f: Y \to Z$ and $g: X \to Y$, the composite function $(f \circ g): X \to Z$ is defined by:

The natural domain of $f \circ g$ consists of all $x$ in the domain of $g$ whose outputs $g(x)$ lie in the domain of $f$:

Note that function composition is generally non-commutative: $f \circ g \ne g \circ f$ in general, though it is always associative: $f \circ (g \circ h) = (f \circ g) \circ h$.

2. Invertibility: Injectivity, Surjectivity & Bijectivity

Theorem (Existence of Inverse Function): A function $f: X \to Y$ possesses a two-sided inverse function $f^{-1}: Y \to X$ if and only if $f$ is a bijection. When $f^{-1}$ exists, it satisfies:

Geometrically, the point $(a, b)$ lies on the graph of $y = f(x)$ if and only if $(b, a)$ lies on the graph of $y = f^{-1}(x)$. Thus, the graph of $f^{-1}$ is the exact reflection of the graph of $f$ across the principal diagonal line $y = x$.