Increasing and decreasing functions — intuitive definition

2.2.11 · Maths › Functions

"Increasing" Ka Matlab Kya Hai?

YEH definition KYUN? Hum "consistently upar jaana" ka idea capture karna chahte hain. Agar do points $x_{1}$ aur $x_{2}$ ordered hain ( $x_{1}$ , $x_{2}$ ke left mein hai), toh unke function values bhi same direction mein ordered hone chahiye ( $f (x_{1})$ , $f (x_{2})$ se neeche hai). Koi dip nahi, koi flat section nahi — pure upar ka trend.

HUM KYA compare kar rahe hain? Steepness nahi (woh baad mein derivative ka kaam hai). Sirf order: left point ka output vs. right point ka output.

Visually CHECK KAISE karein? Ek horizontal line kheeecho. Jaise tum use upar slide karo, har horizontal line graph ko uss interval mein zyada se zyada ek baar cross karni chahiye, aur graph left se right tak upar uthna chahiye.

"Decreasing" Ka Matlab Kya Hai?

Inequality KYUN flip hoti hai? Kyunki "neeche utarna" matlab right point, left point se neeche hai. X ka order hai $x_{1} < x_{2}$ , lekin y ka order ulta hai: $f (x_{1}) > f (x_{2})$ .

Strictly vs. Non-Strictly (Zaroori Nuance)

Upar ki definitions strictly increasing aur strictly decreasing ke liye hain (inequalities strict hain: $<$ aur $>$ , na ki $\leq$ ya $\geq$ ).

Yeh KYUN matter karta hai? Ek constant function jaise $f (x) = 5$ ke liye saare $x_{1}, x_{2}$ par $f (x_{1}) = f (x_{2})$ satisfy hota hai. Yeh strictly increasing NAHI hai (kyunki $f (x_{1})$ , $f (x_{2})$ se $<$ nahi hai), aur strictly decreasing bhi NAHI hai. Yeh flat hai.

Kuch books define karti hain:

Non-decreasing (ya monotone increasing): $x_{1} < x_{2} ⟹ f (x_{1}) \leq f (x_{2})$ (flat sections allow karta hai).
Non-increasing (ya monotone decreasing): $x_{1} < x_{2} ⟹ f (x_{1}) \geq f (x_{2})$ .

Is note mein, hum STRICT versions par focus karenge (intuitive "hamesha upar jaana" ya "hamesha neeche jaana").

Worked Examples

Worked example Example 2: Quadratic Function

$f (x) = x^{2}$ Sawaal: Kya $f (x) = x^{2}$ $R$ par increasing hai ya decreasing?

Solution:

$(- \infty, 0)$ par test karo: $x_{1} = - 2, x_{2} = - 1$ lo (toh $x_{1} < x_{2}$ ).
- $f (- 2) = 4$ , $f (- 1) = 1$ .
- $f (- 2) = 4 > 1 = f (- 1)$ , toh $f (x_{1}) > f (x_{2})$ → $(- \infty, 0)$ par decreasing.
- Yeh step KYUN? Humne ek specific pair check kiya. Lekin definition require karti hai ki yeh saari pairs ke liye hold kare. Kyunki squaring negatives ko magnitude mein bada banaata hai par order flip kar deta hai, yeh saare $x_{1} < x_{2} < 0$ ke liye hold karta hai.
$(0, \infty)$ par test karo: $x_{1} = 1, x_{2} = 2$ lo.
- $f (1) = 1$ , $f (2) = 4$ .
- $f (1) = 1 < 4 = f (2)$ , toh $f (x_{1}) < f (x_{2})$ → $(0, \infty)$ par increasing.
Conclusion: $f (x) = x^{2}$ poore $R$ par na increasing hai na decreasing, lekin yeh hai:
- $(- \infty, 0)$ par strictly decreasing.
- $(0, \infty)$ par strictly increasing.
- Yeh KYUN matter karta hai? Monotonicity (increasing/decreasing behavior) ek interval property hai, saare functions ke liye global property nahi hai.

Common Mistakes

Active Recall Practice

Recall Feynman Explanation (12 saal ke bacche ko explain karo)

Socho tum ek pahaad par chadh rahe ho. Agar har ek kadam aage bhi tumhe upar le jaata hai, tum increasing ja rahe ho — raasta upar chadh raha hai. Agar har ek kadam aage tumhe neeche le jaata hai, tum decreasing ja rahe ho — raasta neeche utar raha hai.

Ab, ek function sirf ek rule hai jo ek number (x) leta hai aur tumhe ek aur number (y) deta hai. Agar hum maane ki x kitni door chale ho aur y tumhari height hai, toh:

Increasing function: Aage chalo → tum upar ho.
Decreasing function: Aage chalo → tum neeche ho. Jaise, $f (x) = 2 x$ increasing hai kyunki agar tum bada x choose karo, tumhe bada output milta hai. Agar $x = 1$ , tumhe $f (1) = 2$ milta hai. Agar $x = 3$ , tumhe $f (3) = 6$ milta hai, jo upar hai. Har baar jab tum right jaate ho (bada x), tum upar jaate ho (bada y).

Lekin $f (x) = - x$ decreasing hai. Agar $x = 1$ , $f (1) = - 1$ . Agar $x = 3$ , $f (3) = - 3$ , jo neeche hai (zyada negative). Right jaana matlab neeche jaana.

Kuch functions tricky hote hain. $f (x) = x^{2}$ neeche jaata hai jab x negative ho (jaise -3 se -1 tak, outputs 9 se 1 jaate hain), phir upar jaata hai jab x positive ho. Toh yeh hamesha ek ya dusra nahi hota — depend karta hai ki tum x-axis ke kis part ko dekh rahe ho.

Connections

Monotone functions — broader category (non-strict versions include karta hai)
Derivative and sign of f'(x) — baad mein, increasing test karne ke liye $f^{'} (x) > 0$ use karenge
Intervals and domain — increasing/decreasing specific intervals par define hoti hai
Inverse functions — strictly increasing/decreasing functions ke inverses hote hain
Graphical analysis — increasing/decreasing behavior ka visual inspection
Inequality solving — $f (x_{1}) < f (x_{2})$ check karna aksar algebraic inequalities involve karta hai

#flashcards/maths

Kisi function ka ek interval par increasing hona matlab kya hai?

Interval mein kisi bhi do points

x_{1} < x_{2}

ke liye,

f (x_{1}) < f (x_{2})

hota hai — bada input, bada output deta hai.

Kisi function ka ek interval par decreasing hona matlab kya hai?

Interval mein kisi bhi do points

x_{1} < x_{2}

ke liye,

f (x_{1}) > f (x_{2})

hota hai — bada input, chhota output deta hai.

Kya

f (x) = 3 x - 5

R

par increasing hai ya decreasing?

Increasing.

x_{1} < x_{2}

ke liye,

f (x_{2}) - f (x_{1}) = 3 (x_{2} - x_{1}) > 0

, toh

f (x_{2}) > f (x_{1})

Kya

f (x) = - 2 x + 7

increasing hai ya decreasing?

Decreasing.

x_{1} < x_{2}

ke liye,

f (x_{2}) - f (x_{1}) = - 2 (x_{2} - x_{1}) < 0

, toh

f (x_{2}) < f (x_{1})

Kis interval par

f (x) = x^{2}

increasing hai?

(0, \infty)

(ya

[0, \infty)

agar turning point include karein).

0 < x_{1} < x_{2}

ke liye,

x_{1}^{2} < x_{2}^{2}

Kis interval par

f (x) = x^{2}

decreasing hai?

(- \infty, 0)

(ya

(- \infty, 0]

x_{1} < x_{2} < 0

ke liye,

x_{1}^{2} > x_{2}^{2}

(squaring negatives ke liye order flip kar deti hai).

Kya ek function ek hi interval par ek saath increasing aur decreasing ho sakta hai?

Nahi (jab tak single point na ho). Agar

f

strictly increasing hai, toh

x_{1} < x_{2}

ke liye

f (x_{1}) < f (x_{2})

hoga, jo

f (x_{1}) > f (x_{2})

(decreasing) se contradict karta hai.

"Strictly increasing" aur "non-decreasing" mein kya difference hai?

Strictly increasing:

x_{1} < x_{2} ⟹ f (x_{1}) < f (x_{2})

(strict inequality). Non-decreasing:

x_{1} < x_{2} ⟹ f (x_{1}) \leq f (x_{2})

(flat sections allow karta hai jahan

f (x_{1}) = f (x_{2})

Sach ya Jhooth: Ek constant function

f (x) = c

increasing hai.

Jhooth.

x_{1} < x_{2}

ke liye,

f (x_{1}) = f (x_{2}) = c

, toh

f (x_{1})

f (x_{2})

se kam NAHI hai. Yeh na strictly increasing hai na strictly decreasing.