Concept of a function — input, output, mapping

Why functions matter: They're the language of relationships in mathematics. Temperature depends on time, distance depends on speed, profit depends on sales — all functions. Without this concept, we can't model the universe.

What IS a Function? (Precise Definition)

We write: $f: A \to B$ or $y = f(x)$

• Domain ($A$): The set of all possible inputs (what you're allowed to put in)
• Codomain ($B$): The set where outputs live (the "target" set)
• Range: The set of actual outputs produced (subset of codomain)
• Image of $x$: The output $f(x)$ when we input $x$

WHY "exactly one"? If $f(2)$ could be both 5 and 7, we couldn't predict anything. Mathematics needs determinism. This rule makes functions well-defined.

How Does Mapping Work? The Rule

A function is fundamentally a mapping or correspondence.

The Three Parts:

1. Input set (Domain): Where we choose $x$ from
2. Rule: The operation $f$ that transforms $x$
3. Output set (Codomain/Range): Where $f(x)$ lands

Read as: "f is a function from the real numbers to the real numbers, defined by the rule $f(x) = x^2$"

• $f$ is the name of the function
• $\mathbb{R} \to \mathbb{R}$ specifies domain and codomain
• $x$ is the independent variable (input)
• $f(x)$ is the dependent variable (output)
• $x^2$ is the rule or formula

WHY this notation? It separates the function (the machine) from its inputs. $f$ and $f(x)$ are different: $f$ is the whole process, $f(x)$ is the result of applying it to $x$.

Derivation: Why "Exactly One Output"?

Let's build this from scratch. Suppose we have two sets $A = \{1, 2, 3\}$ and $B = \{4, 5, 6, 7\}$.

Attempt 1 — Is this a function?

• $1 \to 4$
• $2 \to 5$
• $3 \to 6$

Check: Each input (1, 2, 3) has exactly one output. ✓ This IS a function.

Attempt 2 — Is this a function?

• $1 \to 4$
• $2 \to 5$
• $2 \to 7$
• $3 \to 6$

Check: Input 2 maps to both 5 AND 7. ✗ This is NOT a function (it's a relation, but not a function).

WHY does this matter? If we write $f(2) = ?$, we need a single answer. Ambiguity breaks computation.

Attempt 3 — Is this a function?

• $1 \to 4$
• $3 \to 6$

Check: Input 2 has no output. ✗ NOT a function — domain must be fully covered.

Derivation: A vertical line $x = c$ represents all points $(c, y)$ for different $y$ values. If it crosses twice, say at $(c, y_1)$ and $(c, y_2)$, then input $c$ produces two outputs — violating the definition.

Worked Examples — Building Intuition

Compute specific values:

• $f(3) = 3^2 = 9$ — Input 3, output 9
• $f(-2) = (-2)^2 = 4$ — Input -2, output 4
• $f(0) = 0^2 = 0$ — Input 0, output 0

Key observations:

• Domain: All real numbers (any $x$ works)
• Codomain: All real numbers (stated in definition)
• Range: $[0, \infty)$ — outputs are never negative (WHY? A square is always ≥0)

Why this step? We squared because that's the rule. Notice $f(3) \neq f(-3)$ in value, but $f(3) = f(-3) = 9$ — different inputs can give the same output. That's allowed! What's forbidden is one input giving multiple outputs.

The rule: $g(x) = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

Compute:

• $g(5) = 5$ (since $5 \geq 0$, use top branch)
• $g(-5) = -(-5) = 5$ (since $-5 < 0$, use bottom branch)

WHY piecewise? Some functions need different rules for different inputs. Still one output per input! For $x = -5$, we use the second rule (not both), getting exactly 5.

Test: At $x = 0$, solve $0+ y^2 = 1 \Rightarrow y = \pm 1$.

Input $x = 0$ produces two outputs: $y = 1$ and $y = -1$. ✗ Not a function.

How to fix? Restrict to top half: $y = \sqrt{1 - x^2}$ (single-valued). Now it's a function.

Common Mistakes — Steel-Manning Wrong Ideas

Why it feels right: The $\pm$ symbol appears in algebra (quadratic formula), so it seems natural.

The fix: Functions require determinism. If you want both roots, define two functions: $f(x) = +\sqrt{x}$ and $g(x) = -\sqrt{x}$, or return a set (but that's a different type of object, not a standard real function).

Why it feels right: Most early examples are formulas.

The fix: Functions can be defined by:

• Formulas: $f(x) = x^3$
• Tables: $f = \{(1,2), (2,4), (3,6)\}$
• Graphs: A ploted curve
• Words: "Round to nearest integer"
• Algorithms: Computer code

The essence is the input-output pairing, not the representation.

Why it feels right: We see outputs are $\geq 0$ and mix it up.

The fix:

• Domain = inputs you put in
• Range = outputs you get out

For $f(x) = x^2$: Domain $= \mathbb{R}$ (you can square any real), Range $= [0, \infty)$ (squares are non-negative).

Active Recall Practice

That's a function! The numbers you can drop in are the "domain" (like, you can't drop in a basketball, only numbers the box accepts). The numbers that can pop out are in the "codomain." The actual numbers that DO pop out are the "range."

Why do we care? Because everything in the world works like this: time goes in, temperature comes out. Money goes in, stuff comes out. Functions let us predict and understand patterns.

Or: "Function = Faithful Delivery" — mail service analogy. Address (input) → Package (output). One address can't receive two different packages labeled the same way.

Connections to Other Concepts

• Domain and Range: Natural next step — how to find them
• Types of Functions: One-one, onto, bijective classifications
• Inverse Functions: When can we reverse a function?
• Composition of Functions: Chaining functions together
• Relations: Functions are special relations (single-valued)
• Graphs of Functions: Visual representation of mappings
• Real-world Applications: Physics laws, economics, computer science

#flashcards/maths

What is a function? :: A rule that assigns to each input exactly one output. Written $f: A \to B$ where $A$ is domain, $B$ is codomain.

Give a relation that is NOT a function :: The circle $x^2 + y^2 = 1$. At $x=0$, $y = \pm 1$ (two outputs). Any relation where $x$ maps to multiple $y$ values.

Concept Map

Hinglish (regional understanding)