Special Functions and their Graphs
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Friends maine apne previous article mein aapko 'function' se introduce karwaya tha. Uss article mein hamne seekha tha - What is Function?, What is Domain?, What is Codomain?, What is Range? etc.
Agar aapne mera wo article nahi padha hai to pehle uss article ko padhiye kyuki ye waala article aap tab hi sahi se samajh paayenge jab aapne wo waala article padha ho. Mera wo article padhne ke liye diye gaye iss blue link par click kijiye >>>
What is a Function?
Chaliye ab aage badhte hain.
1. Identity function: Maan lijiye ki R, real numbers ka set hai. Tab function f : R → R, f(x) = x ko 'identity function' kehte hain, jaha par x ∈ R. Yaha par iss function ka domain aur range dono hi R hain. Iss function ka graph ek straight line hota hai aur ye line origin se hokar jaati hai.
2. Constant function: Function f : R → R, f(x) = c ko 'constant function' kehte hain jaha par x ∈ R aur c ek constant (fixed number jaise 4 ya -3 ya 2.56 etc.) hai. Iss function ka domain R hai aur range set {c} hai. Iss function ka graph ek straight line hoti hai jo ki X-axis ke parallel hoti hai. For example, f : R → R, f(x) = 3 ka graph aisa hoga.
3. Polynomial function: Ek function f : R → R, f(x) = a0 + a1x + a2x2 + ... + anxn ko 'polynomial function' kehte hain jaha par x ∈ R aur a0, a1, a2, ... , an ∈ R aur n ek non-negative integer hai. Agar short mein kaha jaaye to koi function jo R se R tak ho aur uska rule ek polynomial ho to uss function ko 'polynomial function' kehte hain. Inn functions ke diagram condition ke according alag-alag (different) shape ke bante hain. For example, neeche(below) functions f : R → R, f(x) = x2 aur
f : R → R, f(x) = x3 ke diagrams diye huye hain.
Aap dekh sakte hain ki inn dono functions ke diagram alag-alag (different) hain.
4. Rational functions: Koi function agar iss type ka ho- f : D → C, h(x) aur h(x) = f(x)/g(x) ho. To iss type ke functions ko 'rational function' kehte hain. Yaha par D ka matlab hai function ka Domain aur C ka matlab hai function ka Codomain. Aur f(x) and g(x) dono Domain D mein hi x ke functions hain. Iss type ke functions ke diagrams bhi condition ke according alag-alag (different) shapes ke bante hain. Example ke liye neeche(below) f : R - {0} → R, f(x) = 1/x ka diagram diya gaya hai.
5. Modulus function: Function f : R → R, f(x) = |x| ko 'modulus function' kehte hain jaha par
x ∈ R. |x| ka matlab hai ki agar x ek positive real number hai to |x| ki value wahi positive real number hogi. Aur agar x ek negative real number hai to |x| wahi negative real number hoga lekin uske aage negative sign aayega jisse ki wo negative real number positive real number bann jaayega. Kehne ka matlab ye hai ki agar x negative real number hai to uska negative sign hata(remove) dene ke baad jo number aayega wo |x| ki value hogi. Aur agar x, 0 hai (0 naa hi positive hai aur naa hi negative) to |x| ki value bhi 0 hogi. Isiliye |x| ko aise bhi likha jaata hai,
\[\left | x \right |=\begin{cases} x & \text{, } x\geq 0 \\ -x & \text{, } x< 0 \end{cases}\]
6. Signum function: Function f : R → R, f(x) = |x|/x ko 'signum function' kehte hain. Signum function ko hum aise bhi likhte hain- \[f:R\rightarrow R, f(x)=\begin{cases} 1, & \text{if } x>0 \\ 0, & \text{if } x=0 \\ -1, & \text{if } x<0 \end{cases}\]Isme agar x, 0 se jyada(greater) ho to f(x) = 1 lena hai. Agar x, 0 ho to f(x) = 0 lena hai aur agar x, 0 se kam(less) hai to f(x) = -1 lena hai. Signum function ka domain R hota hai aur range, set {-1, 0, 1} hota hai. Signum function ka graph aisa hota hai-
7. Greatest integer function: Function f : R → R, f(x) = [x] ko 'greatest integer function' kehte hain jaha par x ∈ R. [x] ka matlab hai- 'nearest integer equal to or not greater than x.' For example maan lijiye x, 1.43 hai. Ab 1.43, 1 aur 2 inn dono integers ke beech mein hai but 1, 2 se less hai (matlab greater nahi hai). So [1.43] ki value hogi 1, i.e. [1.43] = 1. Isi tarah se maan lijiye x, -7.3 hai. Ab -7.3, -7 aur -8 ke beech mein hai but -7 aur -8 mein se -8 kam hai (matlab greater nahi hai). So, [-7.3] = -8. Aise hi [3] = 3 and [-1] = -1 etc. Kehne ka matlab ye hai ki x chaahe koi sa bhi real number ho [x] ki value wo integer hogi jo x ke equal ho ya greater naa ho aur x ke nearest (closest) ho. Function
f : R → R, f(x) = [x] ka graph aisa hota hai-
Read Also:
I hope ki mera ye article aap ko pasand aaya hoga. Agar aap ko mera ye article pasand aaya ho to comment karke hame bata sakte hain. Aapke dwaara kiye gaye comment se hame iss tarah ke post/article ko likne ke liye motivation milta hai.
THANKS FOR READING THIS BLOG.
Agar aapne mera wo article nahi padha hai to pehle uss article ko padhiye kyuki ye waala article aap tab hi sahi se samajh paayenge jab aapne wo waala article padha ho. Mera wo article padhne ke liye diye gaye iss blue link par click kijiye >>>
What is a Function?
Chaliye ab aage badhte hain.
Special Functions and their Graphs
1. Identity function: Maan lijiye ki R, real numbers ka set hai. Tab function f : R → R, f(x) = x ko 'identity function' kehte hain, jaha par x ∈ R. Yaha par iss function ka domain aur range dono hi R hain. Iss function ka graph ek straight line hota hai aur ye line origin se hokar jaati hai.
2. Constant function: Function f : R → R, f(x) = c ko 'constant function' kehte hain jaha par x ∈ R aur c ek constant (fixed number jaise 4 ya -3 ya 2.56 etc.) hai. Iss function ka domain R hai aur range set {c} hai. Iss function ka graph ek straight line hoti hai jo ki X-axis ke parallel hoti hai. For example, f : R → R, f(x) = 3 ka graph aisa hoga.
3. Polynomial function: Ek function f : R → R, f(x) = a0 + a1x + a2x2 + ... + anxn ko 'polynomial function' kehte hain jaha par x ∈ R aur a0, a1, a2, ... , an ∈ R aur n ek non-negative integer hai. Agar short mein kaha jaaye to koi function jo R se R tak ho aur uska rule ek polynomial ho to uss function ko 'polynomial function' kehte hain. Inn functions ke diagram condition ke according alag-alag (different) shape ke bante hain. For example, neeche(below) functions f : R → R, f(x) = x2 aur
f : R → R, f(x) = x3 ke diagrams diye huye hain.
Aap dekh sakte hain ki inn dono functions ke diagram alag-alag (different) hain.
4. Rational functions: Koi function agar iss type ka ho- f : D → C, h(x) aur h(x) = f(x)/g(x) ho. To iss type ke functions ko 'rational function' kehte hain. Yaha par D ka matlab hai function ka Domain aur C ka matlab hai function ka Codomain. Aur f(x) and g(x) dono Domain D mein hi x ke functions hain. Iss type ke functions ke diagrams bhi condition ke according alag-alag (different) shapes ke bante hain. Example ke liye neeche(below) f : R - {0} → R, f(x) = 1/x ka diagram diya gaya hai.
5. Modulus function: Function f : R → R, f(x) = |x| ko 'modulus function' kehte hain jaha par
x ∈ R. |x| ka matlab hai ki agar x ek positive real number hai to |x| ki value wahi positive real number hogi. Aur agar x ek negative real number hai to |x| wahi negative real number hoga lekin uske aage negative sign aayega jisse ki wo negative real number positive real number bann jaayega. Kehne ka matlab ye hai ki agar x negative real number hai to uska negative sign hata(remove) dene ke baad jo number aayega wo |x| ki value hogi. Aur agar x, 0 hai (0 naa hi positive hai aur naa hi negative) to |x| ki value bhi 0 hogi. Isiliye |x| ko aise bhi likha jaata hai,
\[\left | x \right |=\begin{cases} x & \text{, } x\geq 0 \\ -x & \text{, } x< 0 \end{cases}\]
Iss tarah se hum modulus function ko aise bhi likhte hain-
\[f:R\rightarrow R, f(x)=\begin{cases} x & \text{, } x\geq 0 \\ -x & \text{, } x< 0 \end{cases}\]
Ye modulus function ko likhne ka ek popular tarika hai. Isme agar x, 0 ya 0 se bada(greater) hai to f(x) ki value x hi hogi aur agar x, 0 se kam(less) hai to f(x) ki value -x hogi.
Modulus function ka graph aisa hota hai-
6. Signum function: Function f : R → R, f(x) = |x|/x ko 'signum function' kehte hain. Signum function ko hum aise bhi likhte hain- \[f:R\rightarrow R, f(x)=\begin{cases} 1, & \text{if } x>0 \\ 0, & \text{if } x=0 \\ -1, & \text{if } x<0 \end{cases}\]Isme agar x, 0 se jyada(greater) ho to f(x) = 1 lena hai. Agar x, 0 ho to f(x) = 0 lena hai aur agar x, 0 se kam(less) hai to f(x) = -1 lena hai. Signum function ka domain R hota hai aur range, set {-1, 0, 1} hota hai. Signum function ka graph aisa hota hai-
7. Greatest integer function: Function f : R → R, f(x) = [x] ko 'greatest integer function' kehte hain jaha par x ∈ R. [x] ka matlab hai- 'nearest integer equal to or not greater than x.' For example maan lijiye x, 1.43 hai. Ab 1.43, 1 aur 2 inn dono integers ke beech mein hai but 1, 2 se less hai (matlab greater nahi hai). So [1.43] ki value hogi 1, i.e. [1.43] = 1. Isi tarah se maan lijiye x, -7.3 hai. Ab -7.3, -7 aur -8 ke beech mein hai but -7 aur -8 mein se -8 kam hai (matlab greater nahi hai). So, [-7.3] = -8. Aise hi [3] = 3 and [-1] = -1 etc. Kehne ka matlab ye hai ki x chaahe koi sa bhi real number ho [x] ki value wo integer hogi jo x ke equal ho ya greater naa ho aur x ke nearest (closest) ho. Function
f : R → R, f(x) = [x] ka graph aisa hota hai-
Read Also:
- What is Set ?
- Types of Set
- Venn Diagrams
- Intersection of Sets
- Difference of Sets
- Symmetric Difference of Sets
- What are Intervals ?
- What is Cartesian Product ?
- What is a Function?
- Coordinate (Cartesian) System - Coordinate Geometry
- Arithmetic Progression (AP)
- How to Solve Quadratic Equation using Quadratic Formula ?
- What is polynomial and equation ?
- What is Euclid's Division Lemma (algorithm) ?
- What is complex number, real number, irrational number, rational number, integer, whole number, natural number and number line?
- What is odd number, even number, prime number, composite number, co-prime number, factor and multiple ?
I hope ki mera ye article aap ko pasand aaya hoga. Agar aap ko mera ye article pasand aaya ho to comment karke hame bata sakte hain. Aapke dwaara kiye gaye comment se hame iss tarah ke post/article ko likne ke liye motivation milta hai.
THANKS FOR READING THIS BLOG.
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