Relations and Functions

Relations and Functions

If to every value of x belonging to some set E there corresponds one or several values of the variable y, then y is called a multiple valued function of x defined on E.Conventionally the word “FUNCTION” is used only as the meaning of a single valued function, if not otherwise stated. Pictorially: \(\xrightarrow [ x ]{ input } \boxed { f }\)
\(\frac{\mathrm{f}(\mathrm{x})=\mathrm{y}}{\text { output }}\) y is called the image of x & x is the pre-image of y under f. Every function from A → B
satisfies the following conditions.

• f ⊂ A x B
• ∀ a ∈ A ⇒ (a, f(a)) ∈ f and
• (a, b) ∈ f & (a, c) ∈ f ⇒ b = c

Domain, Co−Domain & Range Of A Function

Let f : A → B, then the set A is known as the domain of f & the set B is known as co-domain of f. The set of all f images of elements of A is known as the range of f. Thus

Domain of f = { a | a ∈ A, (a, f(a)) ∈ f} Range of f = { f(a) | a ∈ A, f(a) ∈ B}
It should be noted that range is a subset of co−domain. If only the rule of function is given then the domain of the function is the set of those real numbers, where function is defined. For a continuous function, the interval from minimum to maximum value of a function gives the range.

Important Types Of Functions

• Polynomial Function:
  If a function f is defined by f (x) = a₀xⁿ + a₁x^n-1 + a₂x^n-2 + … + a_n-1x + a_n where n is a non negative integer and a₀, a₁, a₂, …, a_n are real numbers and a₀ ≠ 0, then f is called a polynomial function of degree n
  Note:
  (a) A polynomial of degree one with no constant term is called an odd linear function. i.e. f(x) = ax , a ≠ 0
  (b) There are two polynomial functions, satisfying the relation; f(x).f(1/x) = f(x) + f(1/x). They are:
  (i) f(x) = xⁿ + 1 &
  (ii) f(x) = 1 − xⁿ , where n is a positive integer.
• Algebraic Function: y is an algebraic function of x if it is a function that satisfies an algebraicequation of the form
  P₀(x) yⁿ + P₁ (x) y^n-1 + ……. + P_n-1 (x) y + P_n (x) = 0 Where n is a positive integer and
  P₀(x), P₁(x) ……….. are Polynomials in x.
  e.g. y = |x| is an algebraic function, since it satisfies the equation y² − x² = 0.
  Note that all polynomial functions are Algebraic but not the converse. A function that is not algebraic is called Transcedental Function.
• Fractional Rational Function: A rational function is a function of the form.
  \(y=f(x)\frac {g(x)}{h(x)}\)
  where g (x) & h (x) are polynomials & h (x) ≠ 0.
• Absolute Value Function: A function y = f (x) = |x| is called the absolute value function or Modulus function. It is defined as:
  \(\mathrm{y}=|\mathrm{x}|=\left[ \begin{array}{ll}{\mathrm{x}} & {\text { if } \quad \mathrm{x} \geq 0} \\ {-\mathrm{x}} & {\text { if } \quad \mathrm{x}<0}\end{array}\right.\)
• Exponential Function: A function f(x) = a^x = e^xlna (a > 0 , a ≠ 1, x ∈ R) is called anexponential function. The inverse of the exponential function is called the logarithmic function . i.e. g(x) = log_ax.
  Note that f(x) & g(x) are inverse of each other & their graphs are as shown.
• Signum Function:
  A function y= f (x) = Sgn (x) is defined as follows:
  \(\mathrm{y}=\mathrm{f}(\mathrm{x})=\left[ \begin{array}{ccc}{1} & {\text { for }} & {\mathrm{x}>0} \\ {0} & {\text { for }} & {\mathrm{x}=0} \\ {-1} & {\text { for }} & {\mathrm{x}<0}\end{array}\right.\)
  It is also written as Sgn x = |x|/ x ; x ≠ 0 ; f (0) = 0
• Greatest Integer Or Step-Up Function:
  The function y = f (x) = [x] is called the greatest integer function where [x] denotes the greatest integer less than or equal to x. Note that for:
  − 1 ≤ x < 0 ; [x] = − 1 0 ≤ x < 1 ; [x] = 0
  1 ≤ x < 2 ; [x] = 1 2 ≤ x < 3 ; [x] = 2 and so on .
  Properties of greatest integer function:
  
  • [x] ≤ x < [x] + 1 and x − 1 < [x] ≤ x , 0 ≤ x − [x] < 1
  • [x + m] = [x] + m if m is an integer.
  • [x] + [y] ≤ [x + y] ≤ [x] + [y] + 1
  • [x] + [− x] = 0 if x is an integer = − 1 otherwise.
• Fractional Part Function:
  It is defined as:
  g (x) = { x} = x − [x] .
  e.g. the fractional part of the no. 2.1 is
  2.1− 2 = 0.1 and the fractional part of − 3.7 is 0.3. The period of this function is 1 and graph of this function is as shown.

Domains And Ranges Of Common Function

Equal Or Identical Function | Relations and Functions

Two functions f & g are said to be equal if :

• The domain of f = the domain of g.
• The range of f = the range of g and
• f(x) = g(x) , for every x belonging to their common domain.
  eg. f(x) = \(\frac {1}{x}\) & g(x) = \(\frac{x}{x^{2}}\) are identical functions.

Classification Of Functions

One − One Function (Injective mapping):
A function f : A → B is said to be a one−one function or injective mapping if different elements of A have different f images in B. Thus for x₁, x₂ ∈ A & f(x₁), f(x₂) ∈ B , f(x₁) = f(x₂) ⇔ x₁ = x₂ or x₁ ≠ x₂ ⇔ f(x₁) ≠ f(x₂) .
Diagrammatically an injective mapping can be shown as
Note:

• Any function which is entirely increasing or decreasing in whole domain, then f(x) is one−one.
• If any line parallel to the x-axis cuts the graph of the function at most at one point, then the function is one−one.

Many–one function:
A function f : A → B is said to be a many one function if two or more elements of A have the same f image in B. Thus f : A → B is many one if for; x₁, x₂ ∈ A, f(x₁) = f(x₂) but x₁ ≠ x₂
Diagrammatically a many one mapping can be shown as
Note:

• Any continuous function which has atleast one local maximum or local minimum, then f(x) is many−one. In other words, if a line parallel to x−axis cuts the graph of the function atleast at two points, then f is many−one.
• If a function is one−one, it cannot be many−one and vice versa.

Onto function (Surjective mapping): If the function f : A → B is such that each element in B (co−domain) is the f image of atleast one element in A, then we say that f is a function of A ‘onto’ B. Thus f : A → B is surjective iff ∀ b ∈ B, ∃ some a ∈ A such that f (a) = b.
Diagramatically surjective mapping can be shown as
Note: if range = co−domain, then f(x) is onto.

Into function:
If f : A → B is such that there exists atleast one element in co−domain which is not the image of any element in domain, then f(x) is into.
Diagrammatically into function can be shown as

Note: If a function is onto, it cannot be into and vice versa. A polynomial of degree even will always be into. Thus a function can be one of these four types:

• one−one onto (injective & surjective)
• one−one into (injective but not surjective)
• many−one onto (surjective but not injective)
• many−one into (neither surjective nor injective)

Note:
(i) If f is both injective & surjective, then it is called a Bijective mapping. The bijective functions are also named as invertible, non singular or biuniform functions.
(ii) If a set A contains n distinct elements then the number of different functions defined from A→ A is nn & out of it n ! are one one.
Identity function: The function f: A → A defined by f(x) = x ∀ x ∈ A is called the identity of A and is denoted by IA. It is easy to observe that identity function is a bijection.
Constant function: A function f: A → B is said to be a constant function if every element ofA has the same f image in B. Thus f: A → B; f(x) = c, ∀ x ∈ A, c ∈ B is a constant function. Note that the range of a constant function is a singleton and a constant function may be one-one or many-one, onto or into.

Algebraic Operations On Functions

If f & g are real valued functions of x with domain set A, B respectively, then both f & g are defined in A ∩ B. Now we define f+g, f−g, (f. g) & (f/g) as follows:
\(\left.\begin{array}{ll}{\text { (i) }} & {(\mathrm{f} \pm \mathrm{g})(\mathrm{x})=\mathrm{f}(\mathrm{x}) \pm \mathrm{g}(\mathrm{x})} \\ {\text { (ii) }} & {(\mathrm{f} . \mathrm{g})(\mathrm{x})=\mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})}\end{array}\right] \text { domain in each case is } \mathrm{A} \cap \mathrm{B}\)
(iii) \(\left(\frac{f}{g}\right)(x) = \frac {f(x)}{g(x)}\) domain is { x | x ∈ A ∩ B s . t g(x) ≠ 0}.

Composite Of Uniformly & Non-Uniformly Defined Functions

Let f : A → B & g : B → C be two functions . Then the function gof : A → C defined by (gof) (x) = g (f(x)) ∀ x ∈ A is called the composite of the two functions f & g .
Diagrammatically \(\stackrel{x}{\longrightarrow}[f] \stackrel{f(x)}{\longrightarrow}[g] \longrightarrow g(f(x))\). Thus the image of every x ∈ A under the function gof is the g−image of the f−image of x.
Note that gof is defined only if ∀ x ∈ A, f(x) is an element of the domain of g so that we can take its g-image. Hence for the product gof of two functions f & g, the range of f must be a subset of the domain of g.
Properties Of Composite Functions:
(i) The composite of functions is not commutative i.e. gof ≠ fog .
(ii) The composite of functions is associative i.e. if f, g, h are three functions such that fo (goh) & (fog) oh are defined, then fo (goh) = (fog) oh .
(iii) The composite of two bijections is a bijection i.e. if f & g are two bijections such that gof is defined, then gof is also a bijection.

Homogeneous Functions | Relations and Functions

A function is said to be homogeneous with respect to any set of variables when each of its terms is of the same degree with respect to those variables.
For example 5x² + 3y² − xy is homogeneous in x & y . Symbolically if, f(tx , ty) = tⁿ .f(x , y) then f(x , y) is homogeneous function of degree n.

Bounded Function | Relations and Functions

A function is said to be bounded if |f(x)| ≤ M , where M is a finite quantity.

Implicit & Explicit Function | Relations and Functions

A function defined by an equation not solved for the dependent variable is called an Implicit Function. For eg. the equation x³+y³ = 1 defines y as an implicit function. If y has been expressed in terms of x alone then it is called an Explicit Function.

Inverse of A Function | Relations and Functions

Let f : A → B be a one−one & onto function, then their exists a unique function g : B → A such that f(x) = y ⇔ g(y) = x, ∀ x ∈ A & y ∈ B . Then g is said to be inverse of f. Thus g = f^-1: B → A = { (f(x), x) | (x, f(x)) ∈ f}.

Properties of Inverse Function

• The inverse of a bijection is unique.
• If f : A → B is a bijection & g : B → A is the inverse of f, then fog = IB and gof = I_A , where I_A & I_B are identity functions on the sets A & B respectively.
  Note that the graphs of f & g are the mirror images of each other in the line y = x . As shown in the figure given below a point (x ‘,y ‘) corresponding to y = x² (x >0) changes to (y ‘,x ‘ ) corresponding to y = \(\pm\sqrt {x}\), the changed form of x = \(\sqrt {y}\).
• The inverse of a bijection is also a bijection .
• If f & g two bijections f : A → B , g : B → C then the inverse of gof exists and (gof)−1 = f−1o g−1

Odd and Even Functions | Relations and Functions

If f (−x) = f (x) for all x in the domain of ‘f’ then f is said to be an even function. e.g. f (x) = cos x ; g (x) = x² + 3 . If f (−x) = −f (x) for all x in the domain of‘f’ then fis said to be an odd function. e.g. f (x) = sin x ; g (x) = x³ + x .

Note:

• f (x) − f (−x) = 0 => f (x) is even & f (x) + f (−x) = 0 => f (x) is odd.
• A function may neither be odd nor even.
• Inverse of an even function is not defined.
• Every even function is symmetric about the y−axis & every odd function is symmetric about the origin.
• Every function can be expressed as the sum of an even & an odd function. e.g.
• only function which is defined on the entire number line & is even and odd at the same time is f(x)= 0.
• If f and g both are even or both are odd then the function f.g will be even but if any one of them is odd then f.g will be odd.

Periodic Function | Relations and Functions

A function f(x) is called periodic if there exists a positive number T (T > 0) called the period of the function such that f(x + T) = f(x), for all values of x with in the domain of x e.g. The function sin x & cos x both are periodic over 2π & tan x is periodic over π
Note:

• f (T) = f (0) = f (−T) , where ‘T’ is the period.
• Inverse of a periodic function does not exist.
• Every constant function is always periodic, with no fundamental period.
• If f(x) has a period T & g (x) also has a period T then it does not mean that f(x) + g (x) must have a period T. e.g. f(x) = |sinx| + |cosx|.
• If f(x) has a period p, then \(\frac {1}{f(x)}\text {and} \sqrt {f(x)}\) also has a period p.
• If f(x) has a period T then f(ax+b) has a period T/a (a>0).

General: If x, y are independent variables, then:

• f (xy) = f(x) + f(y) ⇒ f(x) = k ln x or f(x) = 0 .
• f(xy) = f(x) . f(y) ⇒ f(x) = xⁿ , n ∈ R
• f(x + y) = f(x) . f(y) ⇒ f(x) = a^kx.
• f(x + y) = f(x) + f(y) ⇒ f(x) = kx, where k is a constant.