Functions
Elementary Functions and Their Properties
About This Series
The concept of a function is one of the most fundamental ideas in mathematics and underlies all of analysis. This series begins with exponential and logarithmic functions and systematically studies the properties of elementary functions.
These functions appear everywhere — in differentiation and integration, differential equations, physics, engineering, and many other fields.
Learning by Level
Functions are presented in four levels, step by step. Each level offers both explanatory articles that develop the concepts carefully and reading pieces you can enjoy in a more relaxed way.
What a Function Is · Exponentials and Logarithms
- What a function is (mappings, domain, range)
- Exponent rules and logarithm properties
- Exponential and logarithmic functions
- Glossary of elementary functions
Composition, Inverses, Symmetry, Periodicity
- Composite functions
- Inverse functions
- Even and odd functions
- Periodic functions
Limits, Continuity, Inverse Trig, Function Sequences
- Limits and continuity of functions
- Inverse trigonometric functions
- Convex functions
- Sequences of functions and uniform convergence
Analytic Functions, Approximation, Function Spaces
- Analytic functions and smoothness
- The Weierstrass approximation theorem
- Functional equations
- A gateway to function spaces
Onward: Special Functions
Beyond the elementary functions lies the world of special functions — the gamma function, Bessel functions, elliptic functions, and hypergeometric functions. A good way in is to survey the map from the advanced reading piece “A Tour of Special Functions” before diving deeper.
Individual Topics
- Logistic Function — the canonical sigmoid, from logistic regression to the logistic map and chaos
- Softmax Function — probability outputs for multi-class classification: the temperature parameter, Gumbel-Softmax, and links to statistical mechanics
Frequently Asked Questions (FAQ)
Q1. What are elementary functions?
Elementary functions are the collective name for functions obtained from polynomials, rational functions, exponential functions, logarithmic functions, trigonometric functions, and their inverses through finitely many arithmetic operations and compositions. They are widely used in calculus and physics.
Q2. What is the relationship between exponential and logarithmic functions?
The exponential function $y = a^x$ and the logarithmic function $y = \log_a x$ are inverses of each other; that is, $y = a^x \iff x = \log_a y$. Their graphs are symmetric about the line $y = x$, and the domain and range are swapped.
Q3. Where is the logistic function used?
The logistic function $\sigma(x) = \dfrac{1}{1 + e^{-x}}$ is an S-shaped curve with range $(0, 1)$ that is used naturally to represent probabilities. In machine learning it is the output function of logistic regression for binary classification and an activation function in neural networks. The logistic map is also a standard model in chaos theory.
Q4. How are the softmax and logistic functions connected?
The softmax function $\text{softmax}(\mathbf{z})_i = \dfrac{e^{z_i}}{\sum_j e^{z_j}}$ is a generalization of the logistic function to multiple classes. For $K = 2$ classes, $\text{softmax}(z_1, z_2)_1 = \sigma(z_1 - z_2)$, so two-class softmax reduces to the logistic function. See the softmax article for details.