Relations And Functions Class 11 Notes Mathematics Chapter 2 - CBSE

Chapter : 2

What Are Relations And Functions ?

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    Cartesian Product Of Set

    Given two non-empty sets P and Q. The Cartesian product P × Q is the set of all pairs of elements from P and Q. i.e., P × Q = {(p, q) : p∈ P and q ∈Q} If either P or Q is a null set then P × Q will also be empty set, i.e., P × Q = Φ.

    Note

    • Two ordered pairs are equal if and only if the corresponding first elements are equal and the second elements are also equal.
    • If there are P elements in A and q elements in B, then there will be pq elements in A × B i.e. if n(A) = p and n(B) = q then n(A × B) = pq and total number of relations is 2pq.

    Relation

    A relation R from a non-empty set A to a non-empty set B is a subset of Cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.

    Domain

    The set of all first elements of the ordered pairs in A relation R from a set A to set B is called the 'domain' of the relation R.

    Co-domain

    The set of all second elements in a relation R from a set A to set B is called the range of the relation R. The whole set of B is called co-domian of the relation R. Therefore, range C co-domain.

    Function

    A relation from a set A to a set B is said to be function if every element of set A has one and only one image in set B. The function ƒ from A to B is denoted by ƒ : A → B.

    Real Valued Function

    A function which has either R or one of its subsets as its range is called a real valued function.

    Identity function

    Let R be a set of all real numbers. Defined the real valued function ƒ : R → R by y = f(x) = x for each xϵR. Such a function is called the identity function.

    Here domain and range of f are R.

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    Constant function

    Define the function ƒ: R → R by ƒ(x) = c, xϵR

    where C is a constant and each xϵR. Here domian of ƒ is R and its range is {C}.

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    Polynomial function

    A function ƒ : R → R is said to be polynomial
    function if for each x in R, y = ƒ(x) = a0 + a1 x + a1 x2 + ... + an xn , where n is a non-negative integer and a0 , a1 , a1 , ..., an ϵR.

    Rational function

    Rational function are type of 

    $$\frac{f(x)}{g(x)}\space\text{where f(x) and g(x)}$$ are polynomial functions of x defined in a domain where g(x)≠ 0.

    Exponential function

    If a is a positive real number other than unity, than a function that associates each xϵR to ax is called the exponential function. i.e., f : R → R defined by

    f(x) = ax where a > 0 and a ≠ 1.

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    Logarithmic function

    If a > 0 a ≠ 1 then function defined by f(x) = log a x, x > 0 is called the logarithmic function.

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    The modulus function

    The function f : R → R defined by f(x) = |x| for each x ϵ R is called modulus function.

    $$\text{f(x)} =\begin{cases}x,\space x\ge 0\\-x, x\lt 0\end{cases}$$

    Signum function

    The function f : R → R defined by

    $$\text{f(x)} =\begin{cases}1,\space x\ge 0\\0,\space x=0\\-1, x\lt 0\end{cases}$$

    is called the signum function. The domain of signum function is R and range is the set {–1, 0, 1}

    $$\text{f'(x)}=\frac{|x|}{x}$$

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    Greatest integer function

    The function f : R →R defined by f(x) = [x], xϵR assumes the value of the greatest integer, less than or equal to x. Such a function is called the greatest integer function.

    e.g., [x] = – 1 for –1 ≤ x < 0

    [x] = 0 for 0 ≤ x < 1

    [x] = 1 for 1 ≤ x < 2

    Addition Of Two Real Functions

    Let f : x → R and g : X → R be any two real functions, where X ⊂ R, then we defined (f + g) : X → R by (f + g)x = f(x) + g(x) for all x ϵ R.

    Subtraction Of A Real Function From Another

    Let f : X → R and g : X → R be any two real function, where X Ì R, then we defined (f – g) : X → R by (f – g) x = f(x) – g(x) for all x ϵ R.

    Multiplication by a Scalar: Let f : X → R be any real valued function and a be a scalar. Here by scalar, we mean a real number. Then the product af is a function from X to R defined by (af)x = a(f(x), xϵR.

    Multiplication By A Scalar

    Let f : X → R be any real valued function and a be a scalar. Here by scalar, we mean a real number. Then the product af is a function from X to R defined by (af )x = af(x), x ϵ R.

    Multiplication Of Two Real Functions

    Let f : X → R and g : X → R be a function then fg : X → R defined by (fg)x = f(x).g(x), for all x ϵ R.

    Quotient Of Two Real Functions

    Let f and g two real functions defined from X → R where X ⊂ R, then quotient of f and g denoted by

    $$\frac{f}{g}\space\text{function defined by}\\\frac{f}{g}.x =\frac{f(x)}{g(x)}\\\space\text{provided g(x)}≠ 0, x\epsilon R.$$