Concept of a function — input, output, mapping

2.2.1 · Maths › Functions

Functions kyun matter karte hain: Ye mathematics mein relationships ki language hain. Temperature time pe depend karta hai, distance speed pe depend karti hai, profit sales pe depend karta hai — sab functions hain. Is concept ke bina, hum universe ko model nahi kar sakte.

What IS a Function? (Precise Definition)

Hum likhte hain: $f:A\to B$ ya $y=f(x)$

• Domain ($A$): Saare possible inputs ka set (jo tum daalne ke allowed ho)
• Codomain ($B$): Woh set jahan outputs rehte hain ("target" set)
• Range: Actually produce hue outputs ka set (codomain ka subset)
• Image of $x$: Output $f(x)$ jab hum $x$ input karte hain

"Exactly one" kyun? Agar $f(2)$ dono 5 aur 7 ho sakta, toh hum kuch bhi predict nahi kar sakte. Mathematics ko determinism chahiye. Ye rule functions ko well-defined banata hai.

How Does Mapping Work? The Rule

Ek function fundamentally ek mapping ya correspondence hota hai.

Teen Parts:

1. Input set (Domain): Jahan se hum $x$ choose karte hain
2. Rule: Operation $f$ jo $x$ ko transform karta hai
3. Output set (Codomain/Range): Jahan $f(x)$ land karta hai

Padho aise: "f ek function hai real numbers se real numbers tak, jo rule $f(x)=x^2$ se define hota hai"

• $f$ function ka naam hai
• $\mathbb{R}\to \mathbb{R}$ domain aur codomain specify karta hai
• $x$ independent variable hai (input)
• $f(x)$ dependent variable hai (output)
• $x^2$ rule ya formula hai

Ye notation kyun? Ye function (machine) ko uske inputs se alag karta hai. $f$ aur $f(x)$ alag hain: $f$ poora process hai, $f(x)$ usse $x$ pe apply karne ka result hai.

Derivation: "Exactly One Output" Kyun?

Isko scratch se build karte hain. Maano hamare paas do sets hain $A=\{1,2,3\}$ aur $B=\{4,5,6,7\}$.

Attempt 1 — Kya ye ek function hai?

• $1\to 4$
• $2\to 5$
• $3\to 6$

Check: Har input (1, 2, 3) ka exactly ek output hai. ✓ Ye function HAI.

Attempt 2 — Kya ye ek function hai?

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

Check: Input 2 dono 5 AUR 7 pe map karta hai. ✗ Ye function NAHI hai (ye ek relation hai, lekin function nahi).

Ye kyun matter karta hai? Agar hum $f(2)=?$ likhein, toh humein ek single jawab chahiye. Ambiguity computation ko tod deti hai.

Attempt 3 — Kya ye ek function hai?

• $1\to 4$
• $3\to 6$

Check: Input 2 ka koi output nahi hai. ✗ Function NAHI — domain poori tarah covered honi chahiye.

Derivation: Ek vertical line $x=c$ alag-alag $y$ values ke liye saare points $(c,y)$ represent karti hai. Agar woh do baar cross kare, maano $(c,y_1)$ aur $(c,y_2)$ pe, toh input $c$ do outputs produce karta hai — definition violate hoti hai.

Worked Examples — Intuition Banana

Specific values compute karo:

• $f(3)=3^2=9$ — 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: Saare real numbers (koi bhi $x$ kaam karta hai)
• Codomain: Saare real numbers (definition mein stated hai)
• Range: $[0,\infty )$ — outputs kabhi negative nahi hote (kyun? Square hamesha ≥0 hota hai)

Ye step kyun? Humne square kiya kyunki woh rule hai. Notice karo $f(3)\ne f(-3)$ value mein nahi, lekin $f(3)=f(-3)=9$ — alag inputs same output de sakte hain. Ye allowed hai! Jo forbidden hai woh hai ek input multiple outputs dena.

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

Compute:

• $g(5)=5$ (kyunki $5\ge 0$ hai, top branch use karo)
• $g(-5)=-(-5)=5$ (kyunki $-5<0$ hai, bottom branch use karo)

Piecewise kyun? Kuch functions ko alag inputs ke liye alag rules chahiye hote hain. Phir bhi ek input per ek output! $x=-5$ ke liye, hum doosra rule use karte hain (dono nahi), exactly 5 milta hai.

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

Input $x=0$ do outputs produce karta hai: $y=1$ aur $y=-1$. ✗ Function nahi.

Fix kaise karein? Top half tak restrict karo: $y=\sqrt{1-x^2}$ (single-valued). Ab ye ek function hai.

Common Mistakes — Galat Ideas Ko Steel-Manning Karna

Kyun sahi lagta hai: $\pm$ symbol algebra mein aata hai (quadratic formula), toh natural lagta hai.

Fix: Functions ko determinism chahiye. Agar dono roots chahiye, do functions define karo: $f(x)=+\sqrt{x}$ aur $g(x)=-\sqrt{x}$, ya ek set return karo (lekin woh ek alag type ka object hai, standard real function nahi).

Kyun sahi lagta hai: Zyaadatar early examples formulas hote hain.

Fix: Functions define ho sakte hain:

• Formulas se: $f(x)=x^3$
• Tables se: $f=\{(1,2),(2,4),(3,6)\}$
• Graphs se: Ek plotted curve
• Words se: "Nearest integer tak round karo"
• Algorithms se: Computer code

Essence hai input-output pairing, representation nahi.

Kyun sahi lagta hai: Hum dekhte hain outputs $\ge 0$ hain aur dono mix ho jaate hain.

Fix:

• Domain = inputs jo tum daalo
• Range = outputs jo tum pao

$f(x)=x^2$ ke liye: Domain $=\mathbb{R}$ (kisi bhi real ko square kar sakte ho), Range $=[0,\infty )$ (squares non-negative hote hain).

Active Recall Practice

Wahi function hai! Jo numbers tum daal sakte ho woh "domain" hai (jaise, tum basketball nahi daal sakte, sirf woh numbers jo box accept karta hai). Jo numbers nikal sakte hain woh "codomain" mein hain. Jo numbers ACTUALLY nikalte hain woh "range" hai.

Hum kyun care karte hain? Kyunki duniya mein sab kuch aise hi kaam karta hai: time jaata hai, temperature aata hai. Paisa jaata hai, saamaan aata hai. Functions humein patterns predict aur samajhne dete hain.

Ya: "Function = Faithful Delivery" — mail service analogy. Address (input) → Package (output). Ek address do alag packages same label ke saath receive nahi kar sakta.

Connections to Other Concepts

• Domain and Range: Natural next step — unhe kaise find karein
• Types of Functions: One-one, onto, bijective classifications
• Inverse Functions: Ek function ko reverse kab kar sakte hain?
• Composition of Functions: Functions ko chain karna
• Relations: Functions special relations hain (single-valued)
• Graphs of Functions: Mappings ki visual representation
• Real-world Applications: Physics laws, economics, computer science

#flashcards/maths

What is a function? :: Ek rule jo har input ko exactly ek output assign karta hai. Likha jaata hai $f:A\to B$ jahan $A$ domain hai, $B$ codomain hai.

Give a relation that is NOT a function :: Circle $x^2+y^2=1$. $x=0$ pe, $y=\pm 1$ (do outputs). Koi bhi relation jahan $x$ multiple $y$ values pe map kare.

Concept Map