Types — constant, linear, quadratic, polynomial, rational, radical, piecewise

2.2.5 · Maths › Functions

Overview

Functions ko unke algebraic form aur behavior ke basis par classify kiya jaata hai. Function types ko samajhna aapko unke graphs predict karne, sahi solution methods chunne, aur mathematics mein patterns pehchaanne mein help karta hai. Har type ki ek signature structure hoti hai jo uska domain, range, continuity, aur asymptotic behavior determine karti hai.

Figure — Types — constant, linear, quadratic, polynomial, rational, radical, piecewise

1. Constant Functions

Derivation from scratch:

Ek function input $x$ ko output $y$ se map karta hai
Agar mapping rule yeh hai ki "ignore $x$ , hamesha $c$ return karo", to $f (x) = c$
Kyun? Chahe koi bhi $x$ plug karo, equation mein koi $x$ term nahi hai, to output input par depend nahi kar sakta

Properties:

Domain: $R$ (saare real numbers)
Range: ${c}$ (single value)
Graph: horizontal line at $y = c$
Slope: $0$ (koi change nahi)

2. Linear Functions

Derivation from first principles:

Idea se start karo: "output mein change, input mein change ke proportional hai" $\frac{Δ y}{Δ x} = m (constant)$
Is rate ko integrate karo: $Δ y = m \cdot Δ x$
Agar hum point $(0, b)$ se start karein, to $x$ units horizontally move karne ke baad: $y - b = m \cdot (x - 0)$
$y$ ke liye solve karo: $y = m x + b$ ✓

Yeh form kyun?

$m$ : Kitna steep? Positive = rising, negative = falling, zero = horizontal
$b$ : Yeh $y$ -axis ko kahan cross karta hai?

3. Quadratic Functions

Derivation from scratch:

Maano humare paas constant acceleration $2 a$ hai
Velocity: $v (x) = 2 a x + b$ (linear)
Position (velocity ko integrate karo): $f (x) = \int (2 a x + b) d x = a x^{2} + b x + c$
Kyun $x^{2}$ ? Linearly changing rate ko accumulate karne se quadratic milta hai

Standard Forms:

Standard form: $f (x) = a x^{2} + b x + c$
- $c$ : $y$ -intercept
Vertex form: $f (x) = a (x - h)^{2} + k$
- Vertex at $(h, k)$
- Convert kaise karein? Complete the square
Factored form: $f (x) = a (x - r_{1}) (x - r_{2})$
- Roots/zeros at $x = r_{1}, r_{2}$

4. Polynomial Functions

Key Properties by Degree:

Degree 0: Constant ( $a_{0}$ )
Degree 1: Linear ( $a x + b$ )
Degree 2: Quadratic ( $a x^{2} + b x + c$ )
Degree 3: Cubic ( $a x^{3} + b x^{2} + c x + d$ )
Degree $n$ : Up to $n$ roots, $n - 1$ turning points

5. Rational Functions

Domain: Saare real numbers except jahan $q (x) = 0$ .

Types of Asymptotes:

Vertical asymptote $x = a$ par agar $q (a) = 0$ aur $p (a) \neq = 0$
- Function $\pm \infty$ ki taraf approach karta hai jaise $x \to a$
Horizontal asymptote $y = L$ agar $lim_{x \to \infty} f (x) = L$
- $p$ aur $q$ ke degrees se determine hota hai:
  - deg $(p) <$ deg $(q)$ : $y = 0$
  - deg $(p) =$ deg $(q)$ : $y = \frac{a _{n}}{b _{n}}$ (leading coefficients ka ratio)
  - deg $(p) >$ deg $(q)$ : koi horizontal asymptote nahi (oblique ho sakta hai)
Oblique asymptote: agar deg $(p) =$ deg $(q) + 1$ , polynomial division karo

6. Radical Functions

Domain restrictions:

Even roots ( $x$ , $4 x$ , ...): radicand $\geq 0$ hona chahiye (real numbers mein even root ke neeche negative nahi)
Odd roots ( $3 x$ , $5 x$ , ...): koi bhi real number theek hai

Domain ki derivation:

$f (x) = g (x)$ real hone ke liye, humein $g (x) \geq 0$ chahiye
Kyun? $a$ us number $b \geq 0$ ke roop mein defined hai jaise $b^{2} = a$ . Koi real $b$ satisfy nahi karta $b^{2} < 0$ .

7. Piecewise Functions

Important concepts:

Continuity at boundaries: Kya $lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = f (a)$ ?
Evaluating: Dekho kaun sa piece apply hota hai, phir wo formula use karo
Graphing: Har piece ko uske domain par graph karo, jumps/gaps dhundho

Worked example Piecewise Function Examples

Example 1: Absolute Value $f (x) = ∣ x ∣ = {x - x if x \geq 0 if x < 0$

$x = 3$ par: pehla piece use karo, $f (3) = 3$
$x = - 2$ par: doosra piece use karo, $f (- 2) = - (- 2) = 2$
Boundary par $x = 0$ :
Left limit: $lim_{x \to 0^{-}} (- x) = 0$
Right limit: $lim_{x \to 0^{+}} x = 0$
$f (0) = 0$
$x = 0$ par Continuous ✓

Example 2: Tax Bracket Model

0.10x & \text{if } 0 \leq x \leq 10{,}000 \\ 1000 + 0.15(x - 10{,}000) & \text{if } x > 10{,}000 \end{cases}$$ - Income $x = 5000$: $T(5000) = 0.10(5000) = 500$ - Income $x = 20{,}000$: $T(20{,}000) = 1000 + 0.15(10{,}000) = 2500$ - **Kyun $+1000$?** Pehle $\$10k$ par 10\% tax lagta hai jo $1000$ deta hai, phir additional income par 15% tax - **$x = 10{,}000$ par continuity check:** - Left se: $0.10(10{,}000) = 1000$ - Right se: $1000 + 0.15(0) = 1000$ - **Continuous** ✓ **Example 3:** Discontinuous Function $$f(x) = \begin{cases} x^2 & \text{if } x < 2 \\ 2x + 1 & \text{if } x \geq 2 \end{cases}$$ - $x = 2$ par: - Left limit: $\lim_{x \to 2^-} x^2 = 4$ - Right limit: $\lim_{x \to 2^+} (2x+1) = 5$ - $f(2) = 2(2) + 1 = 5$ (doosra piece use karo) - **Jump discontinuity:** left limit $\neq$ right limit

Summary Table

Type	General Form	Key Feature	Domain
Constant	$f (x) = c$	Horizontal line	$R$
Linear	$f (x) = m x + b$	Straight line, constant slope	$R$
Quadratic	$f (x) = a x^{2} + b x + c$	Parabola	$R$
Polynomial	$f (x) = a_{n} x^{n} + \dots + a_{0}$	Smooth curve, degree $n$	$R$
Rational	$f (x) = \frac{p ( x )}{q ( x )}$	Asymptotes	$q (x) \neq = 0$
Radical	$f (x) = n g (x)$	Root function	$g (x) \geq 0$ (even $n$ )
Piecewise	Multiple formulas	Alag intervals par alag rules	Varies by piece

Recall Feynman Technique: 12-Saal-Ke-Bacche Ko Samjhao

Socho tumhare paas ek machine hai jo number leta hai aur doosra number deta hai. Alag machines ke alag rules hote hain: Constant machine: Chahe koi bhi number daalo, hamesha same number nikalti hai. Jaise ek tooti hui vending machine jo sirf chocolate deti hai, chahe tum "chips" press karo.

Linear machine: Input par har ek step aage jaane par, output same amount se upar (ya neeche) jaata hai. Jaise stairs chaddhna—har step tumhe same height upar le jaata hai.

Quadratic machine: Output tezi se tezi se badalta hai (ya dheere dheere). Jaise ek ball jo tum upar phenko—yeh dheeli hoti hai, rukti hai, phir neeche aakar tez hoti hai. U-shape banati hai.

Rational machine: Ek polynomial ko doosre se divide karta hai. Kabhi kabhi yeh pagal ho jaati hai aur infinity tak shoot karti hai (vertical asymptote) jahan neeche zero ho. Jaise pizza zero logon mein divide karna—impossible!

Radical machine: Squaring ka ulta karta hai. Agar tumne kuch square kiya tha aur 9 aaya, radical machine tumhe 3 wapas deti hai. Lekin yeh choosy hai—regular numbers mein negative ka square root nahi le sakte.

Piecewise machine: Multiple personalities hain! Jab input chhota ho to ek rule use karta hai, jab bada ho to alag rule. Jaise ticket prices: bacche kam dete hain, adults zyada.

Connections

M02.01 Function Definition — types classify karne se pehle samajhna ki function kya hota hai
M02.03 Domain and Range — har type ki characteristic domain/range restrictions hoti hain
M02.04 Function Transformations — in mein se kisi bhi type ko shift, stretch, reflect karna
M02.06 Inverse Functions — kuch types (jaise quadratics) ko invertible hone ke liye domain restrictions chahiye
M03.01 Limits — rational functions mein asymptotes ko rigorously define karne ke liye chahiye
M03.02 Continuity — piecewise functions mein aksar boundaries par continuity issues hote hain
M04.01 Derivatives — har function type ke characteristic derivative formulas hote hain
M05.01 Integration — polynomials vs. rational functions ko integrate karne ke liye alag techniques chahiye

#flashcards/maths

What is a constant function? :: Ek function $f (x) = c$ jahan output hamesha same value $c$ hota hai, chahe input kuch bhi ho.

What is the slope of a constant function?

Zero (graph ek horizontal line hai jisme koi rise nahi).

For a linear function

f (x) = m x + b

, what do

m

and

b

represent?

m

slope hai (rate of change) aur

b