Function notation — f(x), g(x) — Hinglish notes

2.2.3 · Maths › Functions

Core Concept

What Each Part Means

f = Function ki identity (jaise kisi insaan ka naam) x = Input placeholder (domain mein koi bhi value ho sakti hai) f(x) = "f of x" ya "f at x" = output value

HOW to read it: f(3) ka matlab hai "function f ko evaluate karo jab x = 3 ho"

Working with Function Notation

Why This Notation is Powerful

Formula Derivation: f(x) composition kyun enable karta hai

Goal: Samjho kaise notation complex operations ko facilitate karta hai.

Proper notation ke bina (purana style):

Rule 1: "number ko square karo aur 1 add karo"
Rule 2: "number ko double karo aur 3 subtract karo"
Rule 1 phir Rule 2 ko input 5 pe apply karo: ???

Problem: Hum clearly express nahi kar sakte "Rule 1 ka output Rule 2 ka input banao"

Function notation ke saath: $f (x) = x^{2} + 1, g (x) = 2 x - 3$ $g (f (5)) = g (5^{2} + 1) = g (26) = 2 (26) - 3 = 49$

Why this works:

f(5) ek value hai (26), sirf ek vague "result" nahi
g() koi bhi input accept karta hai, isliye g(26) valid hai
Nesting unambiguous hai: g(f(x)) ka matlab hai "g of the result of f" Notation ki power: Yeh functions ko algebraic objects mein badal deta hai jo hum combine kar sakte hain, jaise numbers.

Recall Feynman Technique: Ek 12-saal ke bachche ko explain karo

Socho tumhare paas ek toy factory hai. Factory ka ek naam hai — chaliye use "Factory F" bolte hain।

Jab tum factory ko ek number dete ho (jaise 3), woh apne special rule ko follow karti hai aur ek toy bahar nikalti hai. Shayad Factory F ka rule hai "number ko double karo aur 1 add karo," toh agar tum use 3 doge, tumhe milega 2(3) + 1 = 7.

Ab, f(3) ka matlab hai "Factory F kya produce karta hai jab tum use 3 dete ho?" Jawaab hai 7.

Sirf "jawaab 7 hai" kyun nahi bolte? Kyunki shayad aur bhi factories ho sakti hain! Factory G ka alag rule ho sakta hai, jaise "number ko square karo." Toh g(3) = 9.

Notation f(x) hume kehne mein help karta hai "Factory F jo bhi karta hai x ke saath" — bina rule ko baar baar repeat kiye. Jaise "Shop A ka sandwich" vs "Shop B ka sandwich" kehna — same input (bread, fillings), lekin alag recipes, toh alag sandwiches!

Aur jab hum f(x+2) likhte hain, hum keh rahe hain "Factory F ko input x+2 do" — agar x 5 hai, toh hum use 7 dete hain, 5 nahi. Jaise "extra cheese wala sandwich" order karna — tum input ko shop ke banane se pehle modify karte ho, baad mein nahi.

Connections

Functions: Domain and Range — f(x) notation assume karta hai ki x domain mein hai
Composite functions — f(g(x)) functions combine karne ke liye is notation ka use karta hai
Inverse functions — f⁻¹(x) notation ko functions "undo" karne ke liye extend karta hai
Graphing functions — (x, f(x)) pairs plot karna rule ko visualize karta hai
Piecewise functions — Notation extend hoti hai f(x) = {multiple rules} tak
Function transformations — f(x+2), f(x)+2, 2f(x), f(2x) sab alag-alag modify karte hain
Limits and calculus — lim[x→a] f(x) is notation ko foundation ki tarah use karta hai
Parametric equations — x(t), y(t) notation ko coordinate functions pe apply karta hai

#flashcards/maths

f(x) ka kya matlab hai? :: f(x) ka matlab hai "function f ka output jab input x ho" ya "x pe f evaluate karna". f function ka naam hai, x input variable hai, aur f(x) resulting output value hai.

Agar f(x) = 2x + 3 hai, toh f(5) kya hai?

f(5) = 2(5) + 3 = 10 + 3 = 13. Hum rule mein har x ki jagah 5 substitute karte hain.

f aur f(x) mein kya fark hai?

f function khud hai (rule ya machine), jabki f(x) input x pe function ki value hai (ek number). f ek process hai, f(x) us process ka product hai.

Agar g(x) = x² hai, toh g(a+1) kya hai?

g(a+1) = (a+1)² = a² + 2a + 1. Poora expression (a+1) rule mein x ki jagah le leta hai.

Sach ya Jhoot: f(x+2) = f(x) + 2 :: Jhoot! f(x+2) ka matlab hai f ko input (x+2) pe evaluate karo, jabki f(x)+2 ka matlab hai f ko x pe evaluate karo phir 2 add karo. f(x)=x² ke liye, f(x+2)=x²+4x+4 lekin f(x)+2=x²+2, jo alag hain.

Agar f(x) = 2x aur g(x) = x+3 hai, toh kya f(g(2)) = g(f(2)) hai?

Nahi! f(g(2)) = f(5) = 10, jabki g(f(2)) = g(4) = 7. Function composition commutative nahi hoti; order matters.

f(g(x)) ko kaise padhte hain?

"f of g of x" ya "f composed with g at x". Pehle g evaluate karo (andar), phir us result pe f apply karo (bahar). Inside-out kaam karo.

Hum sirf y = 2x + 1 likhne ki bajaye f(x) notation kyun use karte hain?

Function notation hume allow karta hai: (1) multiple functions ko naam dekar compare karo (f vs g), (2) functions clearly compose karo (f(g(x))), (3) rule ke baare mein baat karo bina specific values compute kiye, aur (4) inverses express karo (f⁻¹).