Math Function Types and Properties

Definition of a Function

Let D and C be two non-empty sets. A function f from set D to set C is a rule which associates to each element x∈D a unique element y∈C

The element y, written as y=f(x) is called image of x and x is called preimage of y .the set D is called the domain and C the co-domain of function f which is denoted by f:D→ C.

The set f(D)={f(x):x∈D}is called the range of the function f:D→ C.

Image of an element is unique in the sense that no element of D can have more than one image. However preimage may not be unique i.e., an element of C can have more than one preimage.

Every element in D has as image in C but every element in C may not have preimage in D. i.e., usually f(D) is proper subset of C.

What is a Real Function?

A function which has either R or one of its subsets as its range is called a real values function. Further ,if its domain is also either R or a subset of R ,It is called a real function.

What is one-to-one function?

A function f: D-> C is called one-to-one if distinct elements of D have distinct images in C, i.e.

x≠y => f(x)≠f(y) ∀ x,y∈D.

Or f(x)=f(y)=>x=y ∀ x,y∈D.

What is one-to-one onto function ?

A function f:D→C is said to be one-one and onto if f is both one-one and onto

What is an Identity Function ?

Identify Function: Let R be the set of real numbers. Define the real Function

f:R→R

Such that y = f(x) = x ∀ x∈R

Domain of f = R

Codomain of f = R

Range of f = r

Also x ≠ y

=>f(x) ≠f(y) ∀ x∈

f is called identify function.

F is both one-one and onto.

What is a Constant Function Let R be the set of real numbers and c∈R be a fixed real number.

Define f:R→R

Such that y=f(x)=c ∀ x∈

Domain of f = R

CoDomain of f = R

Range of f={c}

Also we have x≠y still f(x) f(y)

F is called the constant function

F is neither one-one nor onto

What is a Polynomial Function

A function f: R →R is sad to be polynomial function if for each x∈R we have

y=f(x) = a₀ + a₁ x + a₂ x² + ⋯ + a_n xⁿ

Where n is non-negative integer and a₀,a₁,……..,a_n ∈R.

If a₀ = a₂ = a₃ = ⋯= a_n = 0 and a₁ = 1 the polynomial function become identity function.

If a₁ = a₂ =a₃ = ⋯= a_n = 0 the polynomial function become constant function.

f(x) = x² + 2x = 1 is a polynomial function.

f(x) = x^3/2 + 4x is not a polynomial function.

Consider the function

f: R →R.

Such that y = f(x) = x² ∀ x ∈ R

What are Rational Functions?

Define f: R →R.

Such that f(x)=p(x)/q(x) ,q(x) ≠ 0,x ∈ R

Where p(x) and q(x) are polynomials in x.

f(x) is called a rational function.

Consider f(x) = 1/x, x ≠ 0, x ∈ R

Domain of f = R - {0}

Range of f = R - {0}

f:R - {0} → R - {0} is both one-one and onto.

What is a Modulus Function ?

Modulus function

Define f: R →R.

Such that

f is known as modulus function

f(-1)=|-1|=1=|1|=f(1)

Domain of f = R

Range of f={x ∈R∶x≥0}

f is neither one-one nor onto.

The following properties of modulus function cab be easily verified:

• (i) |xy|=|x||y| ∀ x,y ∈R
• (ii) |x|² = x² ∀ x ∈ R
• (iii) |x| ≤c if and only if-c≤x≤c
• (iv) -|x| ≤x ≤|x| ∀ x∈ R
• (v) |x+y| ≤|x|+|y| ∀ x ∈ R
• (vi) |(|x|-|y|)| ≤|x-y| ∀ x, y ∈ R
• (vii) |x-y| ≤|x|+|y| ∀ x,y∈ R

The interval (a+ϵ , a-ϵ) is nothing but ={x ∈R∶|x-a| < ε}

What is a Signum Function

f: R → R.

Such that

Define greatest integer Function

f: R →R.

Such that f(x) = [x] ∀ x ∈ R

=Greatest integer not greater than x