Vertical line test for functions

2.2.4 · Maths › Functions

Definition

Mathematical restatement: Har $a\in \text{domain}$ ke liye, zyada se zyada ek aisa $b$ exist karta hai ki $(a,b)\in R$.

Yeh Kyun Kaam Karta Hai: Function Definition Se Derivation

Function ki formal definition se shuru karte hain: $f:A\to B\text{ is a function }⟺\forall x\in A,\exists !y\in B\text{ such that }y=f(x)$

Symbol $\exists !$ ka matlab hai "exactly ek exist karta hai."

Step 1: Kisi relation ka graph kaisa dikhta hai?

• Graph un saare points $(x,y)$ ka set hota hai jahan relation hold karta hai.
• Ek function $f$ ka graph hai $\{(x,f(x)):x\in \text{domain}\}$.

Step 2: Ek vertical line kya represent karti hai?

• $x=a$ par vertical line yeh set hai $\{(a,y):y\in \mathbb{R}\}$.
• Yeh $x$-coordinate ko fix karti hai aur $y$ ko vary karti hai.

Step 3: Intersection ka matlab kya hai?

• Vertical line $x=a$ graph ko un saare points $(a,y)$ par intersect karti hai jo graph par hain.
• Agar graph ek function represent karta hai, toh har $x=a$ ke liye zyada se zyada ek $y$ ho sakta hai (function ki definition ke according).
• Ek intersection → us $x$ ke liye exactly ek $y$ → function behavior ✓
• Zero intersections → $a$ domain mein nahi → phir bhi theek hai (function wahan simply undefined hai)
• Do ya zyada intersections → $x=a$ ke liye multiple $y$ values → function definition ka violation ✗

Conclusion: Vertical line test, function definition mein uniqueness condition ki ek visual encoding hai.

Worked Examples

Graph: $x^2+y^2=25$ (circle centered at origin, radius 5)

Test: $x=3$ par vertical line kheencho. $x=3⟹9+y^2=25⟹y^2=16⟹y=\pm 4$

Result: Line $(3,4)$ aur $(3,-4)$ par cross karti hai → do points.

Yeh step kyun? Hum check kar rahe hain ki ek $x$ multiple $y$ values deta hai ya nahi. Yahan $x=3$ dono $y=4$ aur $y=-4$ deta hai, toh yeh function nahi hai.

Conclusion: ✗ Function nahi. Yeh relation kai $x$ coordinates par multivalued hai.

Graph: $y=x^2$ (standard parabola opening upward)

Test: Kisi bhi $x=a$ par vertical line kheencho. $x=a⟹y=a^2$

Result: Har $x=a$ ke liye, exactly ek $y=a^2$ hai. Vertical line sirf ek point $(a,a^2)$ par cross karti hai.

Yeh step kyun? Hum verify kar rahe hain ki har input $x$ exactly ek output $y$ produce karta hai. Chahe hum koi bhi $a$ chune, $y=a^2$ unique hai.

Conclusion: ✓ Test pass karta hai → $y=x^2$ ek function hai.

Graph: $y^2=x$ (sideways parabola)

Test: $x=4$ par vertical line kheencho. $x=4⟹y^2=4⟹y=\pm 2$

Result: $(4,2)$ aur $(4,-2)$ par do intersections.

Yeh step kyun? Hum koi bhi aisa $x$ dhundh rahe hain jo multiple $y$ values produce kare. $x=4$ ke liye, hame $y=2$ aur $y=-2$ milta hai, jo function rule ko todta hai.

Conclusion: ✗ Function nahi. (Halaanki $y=\sqrt{x}$ aur $y=-\sqrt{x}$ alag-alag functions hain.)

Graph: $f(x)=\{\begin{array}{ll}x+1& x<0\\ x^2& x\ge 0\end{array}$

Test:

• $x=-2$ ke liye: $y=-2+1=-1$ (ek point)
• $x=0$ ke liye: $y=0^2=0$ (ek point)
• $x=2$ ke liye: $y=2^2=4$ (ek point)

Result: Har vertical line zyada se zyada ek baar intersect karti hai.

Yeh step kyun? Bhale hi rule $x=0$ par change ho jaata hai, phir bhi har $x$ exactly ek $y$ par map karta hai. "Jump" ek single input ke liye multiple outputs nahi banata.

Conclusion: ✓ Pass karta hai → Yeh function hai.

Common Mistakes

Mnemonic

Connections 2.1.01-Definition-of-a-function — Formal definition jise vertical line test visualize karta hai

• 2.2.01-Graph-of-a-function — Functions graphically kaise dikhte hain
• 2.2.05-Horizontal-line-test-for-injective-functions — One-to-one functions ke liye complementary test
• 2.3.02-Domain-and-rangefrom-graphs — Graphs se yeh determine karna ki functions kahan defined hain
• 3.1.01-Inverse-functions — Dono vertical aur horizontal line tests pass karne ki zaroorat hoti hai

#flashcards/maths

Vertical line test kya hai? :: Ek relation tab function hota hai jab aur sirf jab har vertical line uske graph ko zyada se zyada ek point par intersect kare.

Concept Map