Functions, Optimization, and Convexity

Comprehensive vocabulary flashcards covering the definitions, classifications, optimization concepts, and convexity/quasi-convexity properties of scalar and multivariable functions.

Function

Given two sets $A$ and $B$, a rule denoted by $f: A \to B$ that associates to each element of the set $A$ one and only one element of the set $B$ ($\forall x \in A, \exists!\, y \in B : y = f(x)$).

Domain

The set $A$ of inputs on which a function $f: A \to B$ is defined, representing the values that the independent variable can take.

Codomain

The set $B$ assigned as the target for a function's outputs, into which the values produced by the function are intended to fall.

Image (Range)

The collection of all values that a function $f: A \to B$ actually attains when applied to every element of its domain, denoted by $\operatorname{Im} f = f(A) = \{f(x) : x \in A\} \subseteq B$.

Surjective Function

A function $f: A \to B$ satisfying $f(A) = B$, meaning every element in the codomain $B$ is the image of at least one element in the domain $A$.
A surjective function is one whose image (range) equals its entire codomain.

Injective Function

A one-to-one function $f: A \to B$ where distinct elements of $A$ map to distinct images in $B$ ($\forall x_1, x_2 \in A, x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$, or equivalently $f(x) = f(y) \implies x = y$).

Bijective Function

A function $f: A \to B$ that is both surjective and injective.

Cartesian Graph

For a function $f: A \to B$, the set of ordered pairs $\operatorname{Gr} f = \{(x, f(x)) : x \in A\} \subseteq A \times B$, which forms a subset of the plane $\mathbb{R}^2$ for scalar functions.

Vertical Line Test

A geometric test stating that a curve in the plane represents a function if and only if no vertical line intersects the curve more than once.

Horizontal Line Test

A geometric criterion stating that a function is injective (and therefore invertible) if and only if no horizontal line intersects its graph more than once.

Inverse Function

For an injective function $f: A \to B$, the function $f^{-1}: \operatorname{Im} f \to A$ satisfying $f^{-1}(f(x)) = x$ for all $x \in A$ and $f(f^{-1}(y)) = y$ for all $y \in \operatorname{Im} f$, with its graph symmetric to $f$ with respect to the line $y = x$.

Bounded Function

A function $f: A \subseteq \mathbb{R}^n \to \mathbb{R}$ whose image set $\operatorname{Im} f$ is bounded both from above and from below, equivalently characterized by the existence of $k > 0$ such that $|f(x)| \leq k$ for all $x \in A$.

Monotonic Function

A function that is either entirely increasing ($x < y \implies f(x) \leq f(y)$) or entirely decreasing ($x < y \implies f(x) \geq f(y)$) on its domain.

Strictly Increasing Function

A function $f: A \subseteq \mathbb{R} \to \mathbb{R}$ that preserves strict inequalities, such that $x < y \implies f(x) < f(y)$ for all $x, y \in A$.

Order Isomorphism

A property of a strictly increasing function $f: A \subseteq \mathbb{R} \to \mathbb{R}$ characterized by preserving order on $A$, such that $x \geq y \iff f(x) \geq f(y)$ for all $x, y \in A$.

Global Maximizer

A point $\hat{x} \in C \subseteq A$ such that $f(\hat{x}) \geq f(x)$ for all $x \in C$. The corresponding value $f(\hat{x})$ is the unique global maximum value.

Global Minimizer

A point $\hat{x} \in C \subseteq A$ such that $f(\hat{x}) \leq f(x)$ for all $x \in C$. The corresponding value $f(\hat{x})$ is the unique global minimum value.

Solution Set (Argmax)

The set of all global maximizers of an objective function $f$ on a choice set $C$, denoted by $\operatorname{argmax}_{x \in C} f(x) = \{\hat{x} \in C : f(\hat{x}) = \max_{x \in C} f(x)\}$.

Strong Maximizer

A point $\hat{x} \in C$ satisfying $f(\hat{x}) > f(x)$ for all $x \in C$ with $x \neq \hat{x}$. A maximizer is strong if and only if it is unique (i.e., $\operatorname{argmax}_{x \in C} f(x)$ is a singleton).

Local Maximizer

A point $x_0 \in C$ for which there exists an $\epsilon$-neighborhood $B_\epsilon(x_0)$ such that $f(x_0) \geq f(x)$ for all $x \in B_\epsilon(x_0) \cap C$.

Local Minimizer

A point $x_0 \in C$ for which there exists an $\epsilon$-neighborhood $B_\epsilon(x_0)$ such that $f(x_0) \leq f(x)$ for all $x \in B_\epsilon(x_0) \cap C$.

Concave Function

A function $f: C \to \mathbb{R}$ on a convex set $C$ satisfying $f(\alpha x + (1-\alpha)y) \geq \alpha f(x) + (1-\alpha)f(y)$ for all $x, y \in C$ and $\alpha \in [0, 1]$, meaning line segments connecting any two points on its graph lie on or below the graph.

Convex Function

A function $f: C \to \mathbb{R}$ on a convex set $C$ satisfying $f(\alpha x + (1-\alpha)y) \leq \alpha f(x) + (1-\alpha)f(y)$ for all $x, y \in C$ and $\alpha \in [0, 1]$, meaning chords connecting any two points on its graph lie on or above the graph.

Affine Function

A function $f: C \to \mathbb{R}$ on a convex set $C$ that is simultaneously concave and convex, satisfying $f(\alpha x + (1-\alpha)y) = \alpha f(x) + (1-\alpha)f(y)$ for all $x, y \in C$ and $\alpha \in [0, 1]$ (e.g., straight lines or flat planes).

Inflection Point

A point $x_0$ in the domain of a function $f$ that separates a region where $f$ is concave from a region where $f$ is convex, or vice versa.

Fenchel Theorem

A theorem establishing that if $f: C \subseteq \mathbb{R}^n \to \mathbb{R}$ is a concave function defined on a convex set $C$, any local maximizer $\hat{x} \in C$ is also a global maximizer (and conversely, any local minimizer of a convex function on a convex set is a global minimizer).

Level Curve

For a function $f: A \subseteq \mathbb{R}^n \to \mathbb{R}$, the preimage $f^{-1}(k) = \{x \in A : f(x) = k\}$, representing all points in the domain where the function attains the constant value $k \in \mathbb{R}$.

Indifference Curve

A level curve $u^{-1}(k) = \{x \in A : u(x) = k\}$ of a utility function $u: A \subseteq \mathbb{R}_+^n \to \mathbb{R}$, representing baskets of goods that provide the exact same utility level $k$ to a consumer.

Upper Contour Set (Superlevel Set)

For a function $f: A \subseteq \mathbb{R}^n \to \mathbb{R}$, the set $(f \geq k) = \{x \in A : f(x) \geq k\}$ consisting of all domain points whose functional output is at least $k$.

Lower Contour Set (Sublevel Set)

For a function $f: A \subseteq \mathbb{R}^n \to \mathbb{R}$, the set $(f \leq k) = \{x \in A : f(x) \leq k\}$ consisting of all domain points whose functional output is at most $k$.

Quasi-Concave Function

A function $f: C \to \mathbb{R}$ on a convex set $C$ satisfying $f(\alpha x + (1-\alpha)y) \geq \min\{f(x), f(y)\}$ for all $x, y \in C$ and $\alpha \in [0, 1]$, equivalently characterized by every upper contour set $(f \geq k)$ being convex.

Quasi-Convex Function

A function $f: C \to \mathbb{R}$ on a convex set $C$ satisfying $f(\alpha x + (1-\alpha)y) \leq \max\{f(x), f(y)\}$ for all $x, y \in C$ and $\alpha \in [0, 1]$, equivalently characterized by every lower contour set $(f \leq k)$ being convex.

Quasi-Affine Function

A function that is simultaneously quasi-concave and quasi-convex, meaning $\min\{f(x), f(y)\} \leq f(\alpha x + (1-\alpha)y) \leq \max\{f(x), f(y)\}$ for all $x, y \in A$ and $\alpha \in [0, 1]$.

Separable Function

A multivariable function expressible as a sum of functions of independent variables (such as $f(x_1, x_2) = x_1^2 + e^{x_2}$), which preserves convexity or concavity when each component function is convex or concave.