10.36 min read

Functions of One Variable

Before tackling multivariate calculus, we build intuition with functions of one variable $f: \mathbb{R} \to \mathbb{R}$ . These assign a single number $f(x)$ to each real number $x$ . Familiar examples: $f(x) = x^2$ (a parabola), $f(x) = \sin(x)$ , $f(x) = e^x$ .

A function can be visualized as its graph: the set of points $(x, f(x))$ . The graph gives a complete picture of $f$ . Key features: where is $f$ large or small? Is it increasing (going up) or decreasing (going down)? Does it have peaks, troughs, or flat regions? All these can be read from the graph.

Defining properties like limits and derivatives for univariate functions first lets us build intuition before generalizing to $\mathbb{R}^n$ .

Formal View

Definition 10.2 — Univariate Function

A function of one variable is a map

f: D \to \mathbb{R}

where

D \subseteq \mathbb{R}

is the domain. The graph of

f

is the set

\{(x, f(x)) : x \in D\} \subset \mathbb{R}^2

Why This Matters

Understanding univariate functions deeply is the foundation for multivariate calculus.

Loss curves during training: $\mathcal{L}(t)$ as a function of training step $t$ .
Activation functions in neural networks: $\text{ReLU}(x) = \max(0,x)$ , $\sigma(x) = 1/(1+e^{-x})$ .
Interest rate models: $P(t) = P_0 e^{rt}$ as a function of time.

Learning Resources

Functions and their graphs

Khan Academy

Introduction to functions and how to read their graphs.

10 min

Essence of calculus, Chapter 1

3Blue1Brown

Overview of what calculus is about, starting from functions.

17 min

Quiz

Question 1

The graph of $f(x) = x^2$ is:

Question 2

Every subset of $\mathbb{R}^2$ is the graph of a function.

Common Mistakes

Confusing a function $f$ with its graph — the function is the rule; the graph is its visual representation.
Thinking the domain is always all of $\mathbb{R}$ — functions like $f(x) = \sqrt{x}$ have restricted domains.