2nd year math ch 1 Ex 1.2 – Graphical Representaion of Functions – New Book 2026-27

Bunyadi Concept

Sabse pehle ye samajhna zaroori hai ke functions do types ke hote hain:

• Algebraic function: Wo function jo sirf addition, subtraction, multiplication, division, aur root (rational power) se x se banaya ja sake. Jaise: polynomials, rational functions (fractions of polynomials), radical functions (square roots waghera).
• Transcendental function: Wo function jo in simple operations se nahi banaya ja sakta — jaise trigonometric (sin, cos, tan), exponential (eˣ, aˣ), logarithmic, aur hyperbolic functions.

Yaad rakhne ka tareeqa: Agar function mein x sirf power, root, ya fraction ki soorat mein ho → algebraic. Agar x kisi trig, exponential, ya log ke andar ho → transcendental.

Question 1 — Algebraic Functions Pehchano

Panch functions diye the, har ek ko check karna tha:

• sec x → ye trigonometric hai → transcendental (algebraic nahi)
• (2x−3)/(x+1) → ye do polynomials ka ratio hai (fraction) → algebraic
• √(x²+2x) → ye polynomial ka root hai → algebraic
• (eˣ−e⁻ˣ)/2 → ye sinh x hai, exponential se bana hai → transcendental
• (x−1)³ → simple polynomial hai → algebraic

Answer: Sirf teen functions algebraic hain: (2x−3)/(x+1), √(x²+2x), aur (x−1)³

Question 2 — Transcendental Functions Pehchano

Same tareeqa, ulta poocha gaya:

• 1/x → ye rational function hai (fraction) → algebraic (transcendental nahi)
• cot x → trigonometric → transcendental
• e²ˣ + √(x²+2x) → isme e²ˣ mojood hai, isliye poori expression transcendental ban jaati hai (chahe dusra hissa algebraic ho)
• (eˣ+e⁻ˣ)/2 → ye cosh x hai → transcendental
• (x−1)^(−3/2) → ye polynomial ka rational power hai → algebraic (transcendental nahi)

Answer: cot x, e²ˣ+√(x²+2x), aur cosh x transcendental hain.

Important nuqta: Agar kisi expression mein ek bhi transcendental term mojood ho (jaise eˣ), to poori expression transcendental ban jaati hai, chahe baqi hissa algebraic ho.

Question 3 — Fundamental Transcendental Functions

Ab agla level: transcendental functions ko do parts mein baanta hai:

• Fundamental: Basic named function jo seedha x par lagi ho (jaise sin x, eˣ, cosh x)
• Non-fundamental (composite): Jab do alag transcendental functions ko mila diya jaye (jaise eˣ + cos x)
• eˣ + cos x → ye do transcendental functions ka sum hai → non-fundamental
• sin x → basic trig function → fundamental
• e²ˣ → ye (e²)ˣ likha ja sakta hai, yani phir bhi pure aˣ form hai → fundamental
• (eˣ−e⁻ˣ)/2 = sinh x → basic hyperbolic function → fundamental
• 15ˣ → basic exponential form → fundamental

Answer: sin x, e²ˣ, sinh x, aur 15ˣ fundamental hain.

Question 4 — Non-Fundamental Transcendental Functions

• e²ˣ, cos x, 2ˣ, 2⁻ˣ — ye sab basic/pure forms hain → fundamental (is category mein nahi aate)
• cos(x²) → yahan cosine x ka nahi, balke x² ka hai. Ye ek genuine composition hai (do functions ek dusre ke andar) → non-fundamental

Answer: Sirf cos(x²) non-fundamental hai.

Farq samjho: e²ˣ ko fundamental isliye kaha kyunke ise (e²)ˣ likh kar wapis simple aˣ form mein le aa sakte hain. Lekin cos(x²) ko is tarah simplify nahi kar sakte — ye ek function ke andar dusra function hai (composition), isliye non-fundamental hai.

Question 5 aur 6 — Exponential Function ka Domain, Range, Graph

Q5: f(x) = 3ˣ

• Domain: Har real number ke liye 3ˣ defined hai → Domain = (−∞, ∞)
• Range: 3ˣ hamesha positive hota hai (kabhi zero ya negative nahi hota) → Range = (0, ∞)
• Behaviour: Base (3) > 1 hai, isliye function increasing hai (barhta jata hai)
  • x → −∞ hone par f(x) → 0 (x-axis ke qareeb aata hai lekin chhoo nahi paata — ye horizontal asymptote kehlata hai)
  • x → ∞ hone par f(x) → ∞
  • x=0 par f(0)=1, isliye graph hamesha (0,1) se guzarta hai

Q6: f(x) = (1/3)ˣ

• Domain aur range same hain jaise Q5 (Domain = R, Range = (0,∞))
• Lekin base (1/3) < 1 hai, isliye ye function decreasing hai (ulta behaviour)
  • x → −∞ par f(x) → ∞
  • x → ∞ par f(x) → 0

Important concept: (1/3)ˣ asal mein 3⁻ˣ ke barabar hai, isliye iska graph 3ˣ ka mirror image hai y-axis ke around.

Question 7 — Hyperbolic Identities Prove Karna

Ye sari identities definitions se shuru hoti hain:

sinh⁡x=ex−e−x2,cosh⁡x=ex+e−x2\sinh x = \frac{e^x – e^{-x}}{2}, \quad \cosh x = \frac{e^x + e^{-x}}{2}sinhx=2ex−e−x​,coshx=2ex+e−x​

(i) 1 − tanh²x = sech²x
Pehle basic identity nikali: cosh²x − sinh²x = 1 (definitions ko expand karke, terms cancel ho jaate hain aur sirf 1 bachta hai). Phir dono taraf cosh²x se divide kiya to seedha result mil gaya.

(ii) csch²x = coth²x − 1
Same identity (cosh²x − sinh²x = 1) ko is dafa sinh²x se divide kiya.

(iii) sinh 2x = 2 sinh x cosh x
Left side aur right side dono ko alag alag expand kiya (exponential definitions use karke), aur dikhaya ke dono same answer (e²ˣ−e⁻²ˣ)/2 par aa jaate hain.

(iv) cosh 2x = cosh²x + sinh²x
Same tareeqa — right side expand kiya, terms combine hue, aur cosh 2x ki definition mil gayi.

Trick yaad rakho: Har identity prove karne ke liye bas eˣ aur e⁻ˣ ki definitions expand karo, algebra karo, aur terms cancel/combine karke result nikal aata hai.

Question 8 — Inverse Hyperbolic Formulas Prove Karna

Har part mein same standard tareeqa use hua hai:

1. Maan lo y = (inverse function), phir x = (uski normal hyperbolic function) likho
2. Definition ko exponential form mein likho (eʸ aur e⁻ʸ use karke)
3. Isse ek quadratic equation ya simple algebra bante hai eʸ (ya t = eʸ) ke liye
4. Us equation ko solve karo (quadratic formula use karke jahan zaroorat ho)
5. Sahi sign (+ ya −) choose karo (kyunke eʸ hamesha positive hota hai, ya principal value ki wajah se)
6. Aakhir mein dono taraf ln lagao takay y akela reh jaye

Char parts (tanh⁻¹, coth⁻¹, sech⁻¹, csch⁻¹) sab isi pattern se solve hue hain, bas thoda alag algebra ke saath (kabhi seedha linear equation, kabhi quadratic)