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Lesson 01 · Foundations · 6 min

Easy

Functions

Before calculus can describe change, we need a way to describe relationships. A function is just that: a dependable rule that turns each input into exactly one output.

Build the intuition

A rule you can trust

A function is a machine: numbers go in, numbers come out, and the same input always gives the same output. “Square it” is a function. “Add 3” is a function. “Your height at age t” is a function of t.

f(x)=x2f(x) = x^2

Reading f(x) out loud

The notation f(x) reads “f of x” — the output of machine f when you feed it x. So f(3) = 9 just says: put 3 in, get 9 out. The letter x is a placeholder, not a mystery.

Functions are everywhere you look

Temperature throughout the day, the cost of n coffees, battery percentage over time — each is one quantity depending on another. Calculus will ask one question about all of them: how does the output respond when the input changes?

See it move

InteractiveThe function machine
in 1.5
square it x²
out 2.25
1.5
Every input has exactly one output. Plotting all (input, output) pairs draws the curve — a picture of the rule itself.

Slide the input and watch the output respond. The curve is every input–output pair, drawn at once.

A worked example

Evaluate a function

  1. Take the rule

    f(x)=x2+1f(x) = x^2 + 1

  2. Feed it 2 — replace every x with 2:

    f(2)=22+1=5f(2) = 2^2 + 1 = 5

  3. Feed it −2:

    f(−2)=(−2)2+1=5f(-2) = (-2)^2 + 1 = 5

  4. Two different inputs may share an output — but one input never gives two outputs. That reliability is what makes a function a function.

Out in the world

Streaming quality

Your video resolution is a function of bandwidth. The app must answer instantly and consistently — same bandwidth, same quality. Software engineering is full of functions in exactly the calculus sense.

Common confusion, cleared

“f(x) means f times x.”

It's not multiplication — f(x) is the output of rule f at input x. The parentheses here mean “applied to.”

“x has to be “solved for.””

x is just the name of the input slot. Nothing is missing or unknown — the function is a complete rule, waiting for inputs.

Recap

  • A function turns each input into exactly one output.
  • f(x) reads “f of x” — apply rule f to input x.
  • Calculus studies how outputs respond to changing inputs.

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