Functions

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Functions #

Discrete MathDifficulty: ★☆☆☆☆Depth: 0Unlocks: 132

Mappings from inputs to outputs. Domain, range, and function notation.

Interactive Visualization #

⏮◀◀▶▶STEP0.25x1xZOOM

t=0s

Core Concepts #

Key Symbols & Notation #

f: A -> B (function f from set A to set B)f(x) (value of f at input x)

Essential Relationships #

Unlocks (5) #

Limitslvl 2Random Variableslvl 2Recursionlvl 2Relationslvl 2Hash Tableslvl 2

Advanced Learning Details

Graph Position #

6

Depth Cost

132

Fan-Out (ROI)

51

Bottleneck Score

0

Chain Length

Cognitive Load #

6

Atomic Elements

23

Total Elements

L0

Percentile Level

L4

Atomic Level

All Concepts (10) #

Teaching Strategy #

Deep-dive lesson - accessible entry point but dense material. Use worked examples and spaced repetition.

A “function” is the idea behind almost everything in math and computing: you give a rule an input, and it gives you an output. Once you can name inputs, name outputs, and write the rule down clearly, you can build calculus, probability, recursion, and data structures on top of it.

TL;DR:

A function is a mapping that assigns each allowed input exactly one output. The domain is the set of allowed inputs. The range (also called the image) is the set of outputs the function actually produces. We write a function as f(x)f(x)f(x) (“f of x”).

What Is a Function? #

Why we care #

In both math and programming, we constantly model a situation where one thing depends on another:

A function is the cleanest way to describe that dependency.

Core idea (informal) #

A function is a rule that takes an input and returns an output.

But math is picky about one key property:

A function must give exactly one output for each input in its domain.

That means:

First examples #

  1. Doubling: input a number, output twice that number.
  1. Square: input a number, output its square.

A non-example (not a function) #

Suppose we claim a “rule” that maps 1 to both 2 and 3.

This violates the “exactly one output” requirement. So it is not a function.

Function notation #

We often name a function with a letter like fff.

Example:

Let f(x)=2xf(x) = 2xf(x)=2x.

Notice the pattern: the parentheses are not multiplication. f(3)f(3)f(3) is “evaluate fff at 3.”

A gentle checkpoint (before formalism) #

To decide if something is a function, ask:

  1. What are the allowed inputs?

  2. For each allowed input, is there exactly one output?

If yes, it’s a function.

Domain and Range (Image): Naming Inputs and Outputs #

Why these sets matter #

A big source of confusion is that a “rule” is not complete until you know what inputs it is willing to accept.

Example: f(x)=1xf(x) = \frac{1}{x}f(x)=x1​.

So we need vocabulary for “what inputs are allowed” and “what outputs come out.”

Domain #

The domain of a function is the set of inputs you are allowed to plug in.

Examples:

In discrete math and CS, the domain is often a finite or countable set:

Range (also called image) #

The range is the set of outputs the function actually produces when you feed it every input in the domain.

Example 1:

Let f(x)=x2f(x)=x^2f(x)=x2 with domain ℝ.

Example 2:

Let the domain be {−1, 0, 2} and define g(x)=x+1g(x)=x+1g(x)=x+1.

Breathing room: domain vs range in one sentence #

A common subtlety (kept light): range vs “all possible outputs” #

Sometimes people informally say “range” when they mean “all outputs we intend the function to land in.” In more formal math, there is also the codomain (the target set). To keep difficulty at 1/5, we’ll focus on:

If you later see notation like f:A→Bf: A \to Bf:A→B, the set BBB is often the codomain, while the range is the subset of BBB that fff actually hits. You don’t need that distinction yet to use functions correctly day-to-day.

Function tables and mappings (discrete viewpoint) #

In discrete math, a function from a small set can be shown as a table.

Example:

Let domain A = {a,b,c}. Define fff by:

xf(x)
a2
b2
c5

This is a valid function because each input appears once and has exactly one output.

Note: multiple inputs are allowed to share the same output (a and b both map to 2). That is still a function.

Function Notation and Evaluation: Reading and Using f(x) #

Why notation matters #

Function notation is like a tiny language. If you can read and write it, you can:

Defining a function with a formula #

A common way to define a function is with an equation like:

This means: take input xxx, multiply by 2, then add 1.

Evaluate at a specific input:

Defining a function with a description #

You can also define a function in words, especially in CS.

Example:

In math, we might call it ℓ(s)\ell(s)ℓ(s) and write ℓ("cat")=3\ell(\text{"cat"})=3ℓ("cat")=3.

Defining a function by cases (piecewise) #

Sometimes a single formula doesn’t cover the rule cleanly. Then we use a piecewise definition.

Example:

∣x∣={xif x≥0−xif x<0|x| =
\begin{cases}
x & \text{if } x \ge 0 \
-x & \text{if } x < 0
\end{cases}∣x∣={x−x​if x≥0if x<0​

Evaluate:

A quick checkpoint: “exactly one output” with piecewise rules #

A piecewise rule is still a function if:

Example of a problem:

h(x)={0if x≥01if x≥0h(x)=
\begin{cases}
0 & \text{if } x \ge 0 \
1 & \text{if } x \ge 0
\end{cases}h(x)={01​if x≥0if x≥0​

If x = 2, which output should it give—0 or 1? Not a function (as written).

Inputs don’t have to be numbers #

In discrete math, inputs and outputs are often objects.

Examples:

The function idea is the same: each input has one associated output.

(Optional mental model) Functions as machines #

Imagine a machine labeled fff:

If you feed in the same x again, you must get the same output again (for a deterministic function).

Applications and Connections: Where Functions Show Up in CS and Math #

Why this node unlocks so much #

Once you understand “input → output with a rule,” many topics become simple variations:

Below are a few concrete connections.

Limits (preview) #

In calculus, you study what value a function approaches as the input approaches a point.

Even stating a limit requires function language:

You can’t talk about that without knowing what f(x)f(x)f(x) means and what inputs are allowed.

Random variables (preview) #

A random variable is literally a function in the math sense:

Example:

That’s a function from outcomes → {0,1,2}.

Recursion (preview) #

A recursive definition defines a function using itself on smaller inputs.

Example idea:

To read that, you must be comfortable with function evaluation and domain restrictions (e.g., nonnegative integers).

Relations vs functions (preview) #

A relation can connect an input to many outputs.

A function is a special kind of relation that connects each input to exactly one output.

This is why functions are often introduced early in discrete math: they’re “well-behaved” relations.

Hash tables (preview) #

A hash function maps a key (like a string) to a bucket index (like an integer).

Even though collisions exist (different keys can map to the same output), it is still a function because:

Summary table: how to recognize the “function pattern” #

AreaTypical inputTypical outputFunction idea
Algebranumber xnumber f(x)compute a value
Probabilityoutcome ωnumber X(ω)measure an outcome
Algorithmsinstanceanswercompute a result
Data structureskeybucket/valuelookup mapping

Notice the repeating theme: define the domain, ensure one output per input, then use the notation to compute and reason.

Worked Examples (3) #

Evaluate a function and identify domain/range (finite set) #

Let the domain be A = {−1, 0, 2, 4}. Define f(x)=x2+1f(x)=x^2+1f(x)=x2+1 for x ∈ A.

  1. Compute f(x) for each x in A.

  2. State the range (set of outputs).

  1. Compute each output:

    • •f(−1) = (−1)² + 1 = 1 + 1 = 2
    • •f(0) = 0² + 1 = 0 + 1 = 1
    • •f(2) = 2² + 1 = 4 + 1 = 5
    • •f(4) = 4² + 1 = 16 + 1 = 17

  2. Collect outputs into a set (remove duplicates if any):

    Range = {2, 1, 5, 17} = {1,2,5,17}

Insight: On a finite domain, the range is easy: evaluate at every input and list the outputs. The range is a set, so duplicates collapse to one element.

Decide whether a mapping is a function #

Consider the mapping from A = {a,b,c} to numbers defined by the pairs:

(a, 1), (b, 1), (c, 2).

Then consider a second mapping defined by:

(a, 1), (a, 2), (c, 2).

For each mapping, decide if it is a function from A to numbers.

  1. Mapping 1: (a,1), (b,1), (c,2)

    Check each input in A:

    • •a appears once → one output (1)
    • •b appears once → one output (1)
    • •c appears once → one output (2)

    So each input has exactly one output ⇒ this IS a function.

  2. Mapping 2: (a,1), (a,2), (c,2)

    Check each input in A:

    • •a appears twice with two different outputs (1 and 2) ⇒ violates “exactly one output”

    Therefore this is NOT a function as a mapping from A.

    (Also, b has no output listed, so even “at least one output” fails if we require the whole domain A.)

Insight: Two common ways to fail being a function: (1) one input maps to two outputs, or (2) an input in the declared domain has no output at all.

Domain restriction example: $f(x)=\frac{1}{x}$ #

Let f(x)=1xf(x)=\frac{1}{x}f(x)=x1​.

  1. Explain why x = 0 cannot be in the domain (over real numbers).

  2. Compute f(2), f(−4), and describe the range over the domain ℝ \ {0}.

  1. Check x = 0:

    f(0) = 1/0 is undefined (division by zero). So 0 must be excluded from the domain.

    Domain (over ℝ) is {x ∈ ℝ : x ≠ 0}.

  2. Evaluate:

    • •f(2) = 1/2
    • •f(−4) = −1/4

  3. Describe the range:

    Can the output ever be 0?

    If 1/x = 0, then 1 = 0·x = 0, impossible.

    So 0 is never produced.

    Range is {y ∈ ℝ : y ≠ 0}.

Insight: Domain restrictions often come from “operations that can break,” like division by zero or square roots of negative numbers (over reals). The range can also exclude values for algebraic reasons.

Key Takeaways #

Common Mistakes #

Practice #

easy

Let f(x)=3x−2f(x)=3x-2f(x)=3x−2. Compute f(0), f(2), and f(−1).

Hint: Substitute each input into 3x−2 and simplify.

Show solution

f(0)=3·0−2=−2

f(2)=3·2−2=6−2=4

f(−1)=3·(−1)−2=−3−2=−5

easy

Domain/range practice: Domain A = {1,2,3,4}. Define g(x)g(x)g(x) by the table: g(1)=2, g(2)=2, g(3)=5, g(4)=5. What is the range of g?

Hint: List the outputs and remove duplicates.

Show solution

Outputs are {2,2,5,5}. As a set, the range is {2,5}.

medium

Is it a function? Consider the relation on A = {a,b,c} given by pairs (a,1), (b,2), (b,3), (c,4). Decide whether this is a function from A to numbers.

Hint: Check whether any input has more than one output, and whether every input has at least one output.

Show solution

Input b maps to both 2 and 3, so the mapping gives two outputs for the same input. Therefore it is NOT a function from A.

Connections #

Quality: B (4.2/5)

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