FUNCTION
A relation ƒ from A to B (A,B are two non-empty sets) is said to be a function from A to B if ∀ x∈A there exists a unique element y∈B s.t. (x,y)∈ƒ.It is denoted by ƒ:A→B .
☛ Domain of a function :
☛ Co-domain of a function :
☛ Range of a function :
☛ Types of function :
🌑 One-One or Injective function :
A function ƒ:A→B is said to be one-one or injective if each element in the domain of a function has a distinct image in the co-domain.
► How to check if the given function is injective or not
🌑 Many one function :
A function ƒ:A→B is said to be many one if there are at least two elements in the domain whose images are same.
Example: ƒ:R→R defined by ƒ(x)=|x| is many one.
Note: All even functions are many one.
🌑 Onto function or Surjective function :
A function ƒ:A→B is said to be surjective if ƒ(A)=B i.e. range =co-domain of function.
Note: If ƒ is surjective,each element of B has at least one pre-image.
Example: ƒ:R→R defined by ƒ(x)=x is surjective function.
🌑 Into function :
A function ƒ:A→B is said to be an into function if there exists at least one element in the co-domain B which is not an image of any element in the domain A.
Example: ƒ:R→R defined by ƒ(x)=x2 .
➣ Bijective function :
A function ƒ:A→B is said to be bijective if ƒ is both injective and surjective.
Note: If ƒ is bijective each element of B has exactly one pre-image.
Example: ƒ:[-π/2,π/2]→[-1,1] defined by ƒ(x)=sinx is bijective function.
☛ Even and Odd function :
If ƒ:A→B is real valued function such that ∀ x∈A⇒-x∈A and if ƒ(-x)=ƒ(x) ∀ x∈A then ƒ is said to be an even function.
Example: ƒ(x)=x2 is an even function.
If ƒ:A→B is real valued function such that ∀ x∈A⇒-x∈A and if ƒ(-x)=-ƒ(x) ∀ x∈A then ƒ is said to be an odd function. Example: ƒ(x)=x3 is an odd function.
Note: ● Even functions are symmetric about the Y-axis.
● Odd functions are symmetric about the origin and it is placed either in the first and third quadrant or in the second and fourth quadrant.
● ƒ(x)=0 is the only function which is both even and odd.
● The product of two even functions is an even function.
● The sum and difference of two even functions is an even function.
● The product of two odd functions is an odd function.
● The sum and difference of two odd functions is an odd function.
☛ Some special functions :
🌑 Identity function :
The function ƒ:R→R defined by ƒ(x)=x ∀ x∈R is said to be identity function.
Domain= R
Range= R
➤It is a bijective function.
🌑 Constant function :
The function ƒ:R→R defined by ƒ(x)=c ∀ x∈R is said to be constant function.(where c∈R is a constant)
Domain= R
Range={c}
➤It is a many one into function.
🌑 Modulus function :
The function ƒ:R→R defined by
ƒ(x)=|x|=x, when x≥0
=-x , when x<0
Domain= R
Range= R+ ⋃ {o}
➤It is a many one function.
🌑 Signum function
The function ƒ:R→R defined by `{(f(x) =(|x|/x), when\ x≠0),(\ \ \ \ \ \ \ =0,when \ x=0):}`
Domain= R
Range={-1,0,1}
➤It is a many one function.
Domain= R
Range= R+
➤It is an injective function.
🌑 Logarithmic function :
A function ƒ:R+→R defined by ƒ(x)=logₐx,(a>0 and a≠1) is called a logarithmic function.
Domain= R+
Range= R
➤It is an injective function.
🌑 Greatest integer function :
The function ƒ:R→R defined by ƒ(x)=[x],(where [x] denotes the greatest integer less than or equal to x) is called greatest integer function.
Domain= R
Range= Z
Note: ● [x]≤x< [x]+1 and x-1< [x]≤x
● [n+x]=n+[x], if n∈Z
● [x]+[-x]=0, if x∈Z
=-1, if x∉Z
● [[x]]=[x]
🌑 Fractional part function :
Domain= R
Range=[0,1]
Note: ● 0≤{x}<1
● {x}+{-x}=0, if x∈Z
=1, if x∉Z
● [{x}]=0, {[x]}=0
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