Functions

Synopsis

1. Function
   A relation ‘f’ from a non-empty set A to another non-empty set B is said to be a function if every element of A has a unique image in B.
    
2. Domain, co-domain and range of a function
   • The domain of ‘f’ is the set (of all inputs) A. No two distinct ordered pairs in ‘f’ have the same first element.
   • Every function is a relation but the converse is not true.
   • If f is a function from A to B and (a, b) ε f, then f(a) = b, where ‘b’ is called the image of ‘a’ under f and ‘a’ is called the pre-image of ‘b’ under f.
   • If f: A → B, then A is the domain and B is the co-domain of f.
   • The range of the function is the set of all images (outputs).
   
   
3. Real function
   • A function which has either R or one of its subsets as its range is called the real valued function.
   • Moreover, a real function has the set of real numbers or one of its subsets both as its domain and as its range.
   
   
4. Types of Functions
   • One-one (Injective) function
   A function f: X → Y is said to be one-one or injective function if an element of A is associated to an unique element of set B.
   i.e. f: X → Y is one-one if and only if, for all x₁, x₂ ∈ X, f(x₁) = f(x₂) ⇒ x₁ = x₂
   
   • Many-one function
   A function f: X → Y is said to be many-one if two or more elements of X have the same image in Y.
   
   • Onto (Surjective) Function
   A function f : A → B is an onto function, if for each b ε B, there is at least one a ε A such that f(a) = b, i.e. if every element in B is the image of some element in A, then f is an onto or surjective function.
   For an onto function, range = co-domain.
   
   • Bijective Function
   A function which is both one-one and onto is called a bijective function or a bijection.
   A one-one function defined from a finite set to itself is always onto, but if the set is infinite, then it is not the case.
   
   • Into Function
   The mapping f : A → B is called an into function if there exists at least one element in B having no pre-image in A.
   
   • Identity function
   i. Function f: X → X is an identity function if f(x) = x for each x ε A.
   
   ii. Graph of the identity function is a straight line which makes an angle of 45° with both the x–axis and the y--axis, respectively. All points on this line have their x and y coordinates equal.
   
   • Constant function
   i. A constant function is one which maps each element of the domain to a constant i.e. for
   f: R → R, f(x) = c for every x ε R.
   ii. Domain of this function is R and range is the singleton set {c}, where c is a constant.
   
   iii. Graph of a constant function is a line parallel to the x-axis. The graph lies above the x-axis if the constant c > 0, below the x-axis if the constant c < 0 and is the same as the x-axis if c = 0.
   
   • Polynomial function
   i. A function f: R → R defined as y = f(x) = a₀ + a₁x + a₂x² + …. + a_n xⁿ, where n is a non-negative integer and a₀, a₁, a₂, …a_n ε R is called a polynomial function.
   ii. A linear polynomial function represents a straight line, while a quadratic polynomial function represents a parabola.
   
   • Rational function
   i. Functions of the form , where f(x) and g(x) ≠ 0 are polynomial functions, are called rational functions.
   ii. Domain of rational functions does not include those points where g(x) = 0. For example, the domain of  is R – {2}.
   • Modulus function
   i. The function f: R → R denoted by f(x) = |x| for each x ε R is called modulus function.
   ii. The modulus function is defined as f(x) = x if x ≥ 0 and f(x) = −x if x < 0.
   iii. The graph of a modulus function is above the x-axis as shown in the figure.
   
   • Step or greatest integer function
   i. A function f: R → R denoted by f(x) = [x], x ε R, where [x] represents the value of greatest integer, less than or equal to ‘x’ is called a step or greatest integer function. It is also called a floor function.
   
   • Smallest integer function
   A function f: R → R denoted by f(x) = [x], x ε R where  is the value of the smallest integer, greater than or equal to ‘x’ is called a smallest integer function. It is also known as the ceiling function.
   • Signum function
   i. A function defined by f(x) = , x ≠ 0 and 0 for x = 0 is called signum function.
   ii. The domain of a signum function is R and its range is {−1, 0, 1}.
   
   • Exponential function
   If ‘a’ is a positive real number other than unity, then a function which relates each x ε R to a^x is called the exponential function.
   
   • Logarithmic function
   If a > 0 and a ≠ 1, then the function defined by f(x) = log_a x, x > 0 is called the logarithmic function.
   
   • Reciprocal function
   The function defined by f : R - {0} → R such that,   is called the reciprocal function.
   
   • Square root function
   The function defined by f : R⁺ → R such that, is called the square root function.
   • Square function
   The function defined by f : R → R such that, f(x) = x² is called the square function.
   
   • Cube function
   The function defined by f : R → R such that, f(x) = x³ is called the cube function.
   
   • Cube root function
   The function defined by f : R → R such that,  is called the cube root function.
   
   • Inverse of a function
   i. If f: A → B is a bijective function, then for every x ∈ A, we have a y ∈ B, such that y = f (x). A new function f^–1 from B to A which associates each element y ∈ B to its pre-image x = f^–1(y) ∈ A can be defined and this function f^–1 is called the inverse of function f.
   ii. Method to find f–1 if f is bijective
   Step 1: Put y = f (x).
   Step 2: Solve y = f (x) and express x in terms of y.
   Step 3: The value of x obtained in step 2 gives f^–1(y).
   Step 4: Replace y by x in f^–1(y) to obtain f^–1(x).
   • Even and odd functions
   i. If f (x) is a function of x such that f(–x) = f(x) for every x ∈ domain, then f (x) is an even function of x.
   ii. If f (x) is a function of x such that f(–x) = – f(x) for every x ∈ domain, then f (x) is an odd function of x.
   
   
5. Algebra of Real functions
   Let f : X → R and g: X → R be two real functions, where X ⊂ R. Then 
   • (f + g): X → R is defined as (f + g) (x) = f(x) + g(x), x ε X.
   • (f – g): X → R is defined as (f – g) (x) = f(x) – g(x), x ε X.
   • (f ∙ g): X → R is defined as (f ∙ g) (x) = f(x) ∙ g(x), x ε X.
   • (kf): X → R is defined as (kf) (x) = k (f(x)), x ε X, where k is a real number.
   • 
   • (f + k) (x) = f(x) + k
   • fⁿ (x) = [f(x)]ⁿ
   
   
6. Composition of functions
   Let f : A → B and g : B → C be two functions. Then a function gof : A → C defined by (gof) (x) = g (f (x), for all x ∈ A is called the composition of f and g.