CAT Quant: Functions – Important Formulas and Concepts

Introduction

  • A function is a relation between a set of inputs (domain) and a set of possible outputs (range) with the property that each input is related to exactly one output. Denoted by f: A \( \rightarrow \) B(read as f maps A into B) where A is the domain and B is the co-domain.
  • If (a, b) ∈ f then ‘b’ is called the image of ‘a’ under f and is written as b = f(a) and ‘a’ is called the preimage of ‘b’.
  • Notation – If x is an element of the domain A, then f(x) represents the output or image of x under the function f.
  • Example – Consider a function f that doubles the input value. If f(x) = 2x, then f(3) = 6.
  • Impotant Points –
    • f \( \subseteq \) A \( \times \) B
    • Every element of A has a unique f-image in B.
    • Two or more elements of A can have the same f-image in B.
    • There may be elements in B which are not f-images of any element of A.
    • The number of functions from a set A containing m elements to another set B containing n elements is nm.

Types of Function

  • One-One Function (Injection)-
    • A function f: A \( \rightarrow \) B is called a one-to-one function if distinct elements of A have distinct images in B.
    • If n(A)=m n(B) = n, then the number of one to one functions is \( ^{n}P_{m} \ (n \ge m)\)
    • A one-one function is possible from A to B if n(A) \( \le \) n(B)
  • Many-One Function –
    • A function which is not one-one is called many-one function
  • Onto Function (Surjection) –
    • A function f: A \( \rightarrow \) B is called an onto function if every element of B is an image of at least one element of A i.e. f: A \( \rightarrow \) B is onto, if for each y \( \in \) B there exists x \( \in \) A such that f(x) = y.
    • If f is onto, Range of f = co-domain of f
    • An onto function is possible from A to B if n(A) \( \ge \) n(B).
    • If n (A)=m and n(B)=2 there are \( [ 2^{m}-2 ] \) onto functions from A to B.
    • If n(A)=m, n(B)=n then number of onto functions are : \( n^{m} \ – \ ^{n}C_{1}(n-1)^{m}+^{n}C_{2}(n-2)^{m} \ … + … \ (n \le m) \)
  • Into Function –
    • A function which is not onto is called into function, i.e. f: \( A \rightarrow B \) is into if Range of f is a proper subset of B.
  • Bijection –
    • If a function is both one-one and onto, then it is called a bijective function or bijection.
    • A bijection from A to B is possible if n(A)=n(B)
    • If ‘A’ is a set with n elements, the number of bijections from A to A (or in A) is n!.
  • Constant Function –
    • A function f: A \( \rightarrow \) B is said to be a constant function if f(x)=k for all x \( \in \) A, where k is a fixed element of B. Range of f=(k)
  • Identity Function –
    • The function f: \( A\rightarrow A \) defined by f(x)=x is called the identity function, denoted by IA.
    • For the identity function, Range = Domain. Symbolically, \( I_{A}(x)=x for x \in A.
    • Identity function is a bijection.
  • Inverse Function –
    • If f: \( A \rightarrow B \) be a bijective function then the function (f-1) : \( B \rightarrow A, \ where \ f^{-1}=\{ (b,a)/(a,b) \in f \} \) is called the inverse of the function f.
    • If f: \( A \rightarrow B \) is bijective, \( f^{-1}: B \rightarrow A \) is also bijective
    • If \( f(a)=b \ then \ a=f^{-1}(b) \).
  • Composition of Function –
    • If f: \( A \rightarrow B \) and g: \( B \rightarrow C \) are two functions, then “gof” is a function from A to C such that gof(a) = g(f(a)) for every a ∈ A and is called the composition of f and g, read as “g composition of f”.

Real Function

  • If A is a non-empty subset of R, then a function f: \( A \rightarrow R \) is called a real function.

Types of Real Functions

  • Explicit and Implicit Functions –
    • If the relation between the variables is of the form y=f(x) then y is an explicit function of x. Similarly x is an explicit function of y if x = f(y). For Example – \( x = y_2 + 10 \)
    • A function which is not explicit is called an implicit function and it is of the form f(x,y) = 0. For Example – \( x^{2}+y^{2}=a_2 \)
  • Even and Odd Function –
    • A function f(x) is said to be an even function if f(-x) = f(x) for all x in its domain. For Example – (cos x), (x2 +2).
    • A function is said to be an odd function if f(-x) = -f(x) for every x in its domain. For Example – (x3), (sin x).
    • Only certain functions are either odd or even. the others are neither odd nor even.
  • Polynomial Function –
    • A function of the form f(x)= \( a_0x^{n} + a_1x^{(n-2)} + a_2x^{(n-2)} + …. + a_n \) where \( a_0, a_1, a_{2} …. a_{n} \) are real numbers, \( a_{0} \ne 0 \ and \ n \in N \) is called polynomial function of degree n.
    • The domain of a polynomial function is R.
  • Modulus Function –
    • The Modulus function is defined by $$ f(x)=|x|=\left\{ \begin{array}{rcl} x \ {if} \ x \ge 0 \\ -x \ {if} \ x < 0 \end{array}\right .$$
    • For Example – |7| = 7 : |-7| = 7. Here Domain of f = R , Range of f = {set of all non-negative real numbers}.
  • Greatest Integer Function (Step Function) –
    • f(x) = [x] = the greatest integer less than or equal to x.
    • For Example – [7.5] = 7, [8.5] = 8, [-7.1] = -8. Here Domain of f = R , Range of f = Z(set of all integers).
  • Signum Function –
    • The Signum function f is defined by $$sg(x) = \left\{ \begin{array}{rcl} \frac{|x|}{x} \ {if} \ x \neq 0 \\ 0 \ {if} \ x = 0 \end{array}\right .$$ Domain of f = R Range = (-1, 0, 1}.
  • Exponential function –
    • The function that associates every real number x to ex is called the exponential function i.e., \( f(x) = e^{x} \).
    • Domain of f = R, Range = {set of all positive real numbers, (R+)}.
  • Logarithmic Functions –
    • The function that associates every positive real number x to log x is called logarithmic function.
    • Domain of f = {set of all positive real numbers (R+)} , Range of f = R
  • Square Root Function –
    • A function f(x) defined by f(x) = \( \sqrt{x} \ , \ x \in R^{+} \) , is called the square root function.
    • Domain = Range = \( [0, \infty ] \) = {set of all non-negative real numbers}.
  • Trigonometric Function –
    • Trigonometric functions are mathematical functions of an angle, primarily used to relate the angles of a triangle to the lengths of its sides.
    • The six fundamental trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).
    • We give the domain and range of various trigonometric functions.
FUNCTIONDOMAINRANGE
sin x$$ R $$$$ [-1 , 1] $$
cos x$$ R $$$$ [-1 , 1] $$
tan x$$ R -\{ (2n+1) \frac{\pi}{2} \ | \ n \in Z \}$$$$ R $$
cot x$$ R -\{ (n\pi) \ | \ n \in Z \}$$$$ R $$
sec x$$ R -\{ (2n+1) \frac{\pi}{2} \ | \ n \in Z \}$$$$(- \infty , -1] \cup [1 , \infty) $$
cosec x$$ R -\{ (n\pi) \ | \ n \in Z \}$$$$(- \infty , -1] \cup [1 , \infty) $$

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