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 2^pq.

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.

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}.

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 + ... + a_n xⁿ , where n is a non-negative integer and a₀ , a₁ , a₁ , ..., a_n 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.

Logarithmic function

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

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}

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

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.