Functions — Types aur Graphs

📊 Function — Definition

Function ek rule hai jo har independent variable x ki ek unique dependent variable y deta hai. f: X → Y means f X se Y mein map karta hai.

Domain = x ki valid values ka set (input) Co-domain = y ki saari possible values (declared range) Range = f(x) ki actual values ka set (always ⊆ Co-domain)

📋 Types of Functions

Function	Form	Graph Shape	Example
Constant	f(x) = c	Horizontal line	f(x) = 5
Identity	f(x) = x	45° line through origin	f(x) = x
Linear	f(x) = mx+c	Straight line	f(x) = 2x+3
Polynomial	f(x) = aₙxⁿ+...+a₀	Smooth curve	f(x) = x²−3x+2
Rational	f(x) = p(x)/q(x)	Curve with asymptotes	f(x) = 1/x
Modulus	f(x) = \|x\|	V-shape	\|x\| = x if x≥0, -x if x<0
Exponential	f(x) = aˣ	Exponential growth/decay	f(x) = 2ˣ, eˣ
Logarithm	f(x) = log(x)	Slow growth curve	f(x) = log₁₀(x)
Signum	sgn(x)	-1, 0, or 1	+1 if x>0, 0 if x=0, -1 if x<0
Greatest Integer	⌊x⌋	Step function	⌊3.7⌋=3, ⌊-1.2⌋=-2

Composite Function

(f∘g)(x) = f(g(x)) — pehle g apply karo, phir f ka result pe
Example: f(x)=x², g(x)=x+1 → (f∘g)(x) = f(g(x)) = (x+1)²

📈 Graphical Representation

Key Graph Properties

Domain = x-axis pe jahan graph defined hai
Range = y-axis pe jo values aati hain
Vertical Line Test: Agar koi vertical line sirf 1 point pe cut kare → function hai
Horizontal Line Test: Agar sirf 1 point pe cut kare → one-one function hai

Modulus |x|: V-shape, vertex at (0,0), domain=ℝ, range=[0,∞)

Exponential eˣ: Domain=ℝ, Range=(0,∞), always increasing

ln(x): Domain=(0,∞), Range=ℝ, passes through (1,0)

1/x: Domain=ℝ\{0}, Range=ℝ\{0}, 2 branches