Function Combination Explanation (Question 11)

Correct Answer: Option C: h = 0.5 * (g - f)

Method 1: The "2 Switches" Analogy (Sabse Aasan Tarika)

Math aur graphs ko ek second ke liye side mein rakhte hain. Isko 2 switches (buttons) ki tarah samjho jo ek timeline (left se right, -1.00 se +1.00 tak) par rakhe hain:

• Switch OFF matlab value 0.
• Switch ON matlab value 1.

1. Teeno Graphs ko Left se Right dekho:

• Graph g (Pehla Switch): Yeh shuru mein OFF (0) hai. Jaise hi hum -0.25 par pahunchte hain, yeh ON (1) ho jata hai aur aage hamesha ON rehta hai.
• Graph f (Doosra Switch): Yeh bhi shuru mein OFF (0) hai. Yeh thoda late chalu hota hai—jab hum +0.25 par pahunchte hain, tab yeh ON (1) hota hai.
• Graph h (Target Result): Hum chahte hain ki bulb sirf -0.25 se lekar +0.25 ke beech mein jale (kyunki graph h mein sirf beech mein pahad/box bana hai, baaki dono side 0 hai).

2. Sirf beech wala hissa kaise milega?

• -0.25 se +0.25 ke beech mein: Sirf g ON (1) hai, aur f abhi tak OFF (0) hai.
• +0.25 ke baad: Dono g aur f ON (1 aur 1) hain, par humein graph h mein wapas 0 (OFF) chahiye.

Dono 1 hain aur answer 0 chahiye, iska matlab minus (subtract) karna padega (1 - 1 = 0)!

Lekin f - g karein ya g - f?

• Beech wale hisse mein g = 1 hai aur f = 0 hai.
• Agar g - f karenge, toh 1 - 0 = +1 aayega (Line upar jayegi).
• Agar f - g karenge, toh 0 - 1 = -1 aayega (Line niche chali jayegi, jo galat hai).
• Isliye g - f pakka hai!

3. Yeh 0.5 kahan se aaya?

• Graph f aur g mein sabse upar number dekho (left side axis par): 1.0 likha hai. Matlab unki height 1 hai.
• Ab Graph h mein sabse upar number dekho: wahan 0.5 likha hai! Matlab h ki height aadhi (half) hai.
• Kisi bhi cheez ko aadha karne ke liye 0.5 se multiply karte hain.

Toh hamara final formula ban gaya: h = 0.5 * (g - f)

Method 2: Point Substitution (x = 0.00 Test)

Agar exam mein jaldi check karna ho, toh x-axis par x = 0.00 pick karo aur teeno graphs par uski height (y-value) dekho:

• f(0) = 0 (x = 0.00 par line niche hai)
• g(0) = 1 (x = 0.00 par line upar hai)
• h(0) = 0.5 (x = 0.00 par line top par hai, jiski height 0.5 hai)

Options Check at x = 0.00:

1. A. h = f - g -> 0 - 1 = -1 (Galat)
2. B. h = 0.5 * (f + g) -> 0.5 * (0 + 1) = 0.5 (x = 0 par sahi hai, par x = 0.50 par 0.5 * (1 + 1) = 1 dega jabki h(0.50) = 0 hai, toh Galat)
3. C. h = 0.5 * (g - f) -> 0.5 * (1 - 0) = 0.5 (Bilkul Sahi!)
4. D. h = 0.5 * (f - g) -> 0.5 * (0 - 1) = -0.5 (Galat)

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