Exponential functions aˣ — graphs, properties, asymptote

3.2.1 · Maths › Exponentials & Logarithms

KIYA hai ek exponential function?

WHY restrictions hain?

$a > 0$ : agar $a < 0$ ho, toh $a^{1/2} = a$ real nahi hoga — function mein har jagah holes ho jaate. Isliye negative bases forbidden hain.
$a \neq = 1$ : kyunki $1^{x} = 1$ sabhi $x$ ke liye hota hai, jo sirf ek flat line hai, koi sachi exponential nahi.
Hum allow karte hain $a =$ koi bhi positive number, jaise $2^{x}$ , $1 0^{x}$ , $(1/3)^{x}$ , $e^{x}$ .

HOW define karte hain $a^{x}$ ko sabhi real $x$ ke liye?

Jo pehle se jaante hain wahan se shuru karo aur ek law ko sach rakhne ki demand karke extend karo: index law $a^{m + n} = a^{m} \cdot a^{n} .$

Positive integers: $a^{3} = a \cdot a \cdot a$ . (Repeated multiplication — yahi anchor hai.)
Zero: hume chahiye $a^{0} \cdot a^{n} = a^{0 + n} = a^{n}$ , toh $a^{0} = 1$ . Kyun? Law ko consistent rakhne ke liye.
Negatives: chahiye $a^{- n} \cdot a^{n} = a^{0} = 1$ , toh $a^{- n} = \frac{1}{a ^{n}}$ .
Fractions: chahiye $(a^{1/2})^{2} = a^{1} = a$ , toh $a^{1/2} = a$ . Generally $a^{p / q} = q a^{p}$ .
Irrationals (jaise $a^{2}$ ): gaps ko continuity se bharo — $2$ ko rationals $1.41, 1.414, \dots$ ke beech squeeze karo aur limit lo.

Graph aur uski properties

Figure — Exponential functions aˣ — graphs, properties, asymptote

Formula

$y = a^{x}$ ki key properties

Property	Statement	WHY
Se guzarta hai	$(0, 1)$	Kyunki $a^{0} = 1$ har base ke liye
Doosra anchor	$(1, a)$	Kyunki $a^{1} = a$
Range	$y > 0$ hamesha	Positive base kisi bhi power pe positive rehta hai
Horizontal asymptote	$y = 0$ ( $x$ -axis)	Neeche derivation dekho
Growth ( $a > 1$ )	Increasing, upar curve karta hai	$> 1$ se baar baar multiply karne pe grow hota hai
Decay ( $0 < a < 1$ )	Decreasing, $0$ ki taraf curve karta hai	Fraction se multiply karne pe shrink hota hai
Kabhi nahi choota	$y = 0$ kabhi reach nahi hota	$a^{x} = 0$ ka koi solution nahi

WHY hai $y = 0$ ek asymptote (aur kabhi crossed nahi hota)?

Lo $a > 1$ . Jab $x \to - \infty$ , likho $x = - N$ jahan $N \to + \infty$ : $a^{- N} = \frac{1}{a ^{N}} .$ Kyunki $a > 1$ , $a^{N} \to \infty$ , toh $\frac{1}{a ^{N}} \to 0^{+}$ . Curve $0$ ke arbitrarily close aata jaata hai lekin positive rehta hai — yahi exactly ek asymptote hai. Yeh kabhi $0$ nahi hota kyunki $1/ (finite positive)$ kabhi $0$ nahi hota.

$0 < a < 1$ ke liye same cheez hoti hai jab $x \to + \infty$ (graph bas mirror image hai).

Worked examples

Common mistakes (steel-manned)

Active recall

Recall Answers padhne se pehle try karo

Woh kaun sa point hai jo har $y = a^{x}$ se guzarta hai, aur kyun?
Range $y > 0$ kyun hai?
$2^{x}$ ka asymptote derive karo jab $x \to - \infty$ .
$y = (1/ a)^{x}$ ka $y = a^{x}$ se kya relation hai?

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek magic bacteria hai jo har ghante double hoti hai. Ek se shuru karo. 1 ghante baad: 2, phir 4, 8, 16… yeh pagalon ki tarah tezi se badhta hai — yahi $2^{x}$ hai. Ab time mein ulte jao: ek ghante pehle aadha tha, usse pehle quarter, phir eighth… yeh chhota se chhota hota jaata hai lekin kabhi actually zero nahi hota (hamesha kuch bacteria rehta hai, chahe ek speck hi sahi). Woh "zero tak kabhi nahi pahunchna" ka floor hi asymptote hai. Aur chahe tum kisi number ko double ya triple karo, "zero ghante" pe hamesha 1 se shuru hote ho — isliye har curve $(0, 1)$ se guzarti hai.

Flashcards

Ek exponential function ki general form kya hai aur base restrictions kya hain?

f (x) = a^{x}

jahan

a > 0, a \neq = 1

a^{x}

ke liye

a > 0

kyun hona chahiye?

Negative bases non-real values dete hain jaise

a^{1/2} = a

; positivity sabhi

x

ke liye real rakhti hai.

a \neq = 1

kyun required hai?

1^{x} = 1

ek flat line hai, genuine exponential nahi.

Woh kaun sa point hai jo sabhi

y = a^{x}

graphs share karte hain aur kyun?

(0, 1)

, kyunki

a^{0} = 1

har base ke liye.

Woh coordinate kaun sa hai jo directly base reveal karta hai?

(1, a)

, kyunki

a^{1} = a

a^{x}

ki range kya hai?

y > 0

(strictly positive, kabhi zero nahi).

y = a^{x}

ka horizontal asymptote kya hai?

y = 0

(

x

-axis).

Prove karo ki

2^{x} \to 0

jab

x \to - \infty

2^{- N} = 1/ 2^{N}

; jab

N \to \infty

2^{N} \to \infty

, toh

1/ 2^{N} \to 0^{+}

Curve

y = 0

ko kyun kabhi touch nahi karta?

a^{x} = 1/ a^{N}

ek positive fraction hai; nonzero denominator wala fraction kabhi 0 nahi hota.

y = (1/ a)^{x}

y = a^{x}

se kya relation hai?

(1/ a)^{x} = a^{- x}

a^{x}

y

-axis mein reflection hai (decay = reversed growth).

a^{x}

aur

x^{a}

mein fark?

a^{x}

exponential hai (variable in exponent);

x^{a}

ek power/polynomial hai (variable in base).

y = 2^{x} + 3

ke asymptote ka kya hota hai?

Woh

y = 3

tak shift ho jaata hai; range ban jaati hai

y > 3

; intercept

(0, 4)

Growth vs decay condition?

a > 1

⇒ increasing (growth);

0 < a < 1

⇒ decreasing (decay).

a^{0} = 1

sirf stated nahi balki derivable kyun hai?

a^{0} \cdot a^{n} = a^{0 + n} = a^{n}

se, divide karne pe

a^{0} = 1

milta hai.

Connections

Logarithms as the inverse of exponentials — $a^{x}$ ko $y = x$ mein reflect karo toh $lo g_{a} x$ milta hai.
The number e and natural exponential eˣ — woh special base jahan slope = height.
Index laws — woh algebra jo $a^{x}$ ko sabhi real $x$ ke liye define karta hai.
Exponential growth and decay models — real-world use (population, radioactivity).
Graph transformations — $a^{x}$ pe shifts/reflections apply karna.