NCERT Notes Mathematics for Class 12

                   

Chapter 1:- Relations and Functions

Let A and B be .two non-empty sets, then a function f from set A to set B is a rule whichassociates each element of A to a unique element of B.

It is represented as f: A → B and function is also called mapping.
f : A → B is called a real function, if A and B are subsets of R.

Domain and Codomain of a Real Function

Domain and codomain of a function f is a set of all real numbers x for which f(x) is a real number. Here, set A is domain and set B is codomain.

Range of a Real Function

Range of a real function, f is a set of values f(x) which it attains on the points of its domain.

Classification of Real Functions

Real functions are generally classified under two categories algebraic functions and transcendental functions.

1. Algebraic Functions

Some algebraic functions are given below
(i) Polynomial Functions If a function y = f(x) is given by

                                                              where, a₀, a₁, a₂,…, a_n are real numbers and n is any non -negative integer, then f (x) is called a polynomial function in x.

If a₀ ≠ 0, then the degree of the polynomial f(x) is n. The domain of a polynomial function is the set of real number R. 

e.g., y = f(x) = 3x⁵ – 4x² – 2x +1

is a polynomial of degree 5.

(ii) Rational Functions If a function y = f(x) is given by f(x) = φ(x) / Ψ(x)

where, φ(x) and Ψ(x) are polynomial functions, then f(x) is called rational function in x.

(iii) Irrational Functions The algebraic functions containing one or more terms having nonintegral rational power x are called irrational functions.
e.g., y = f(x) = 2√x –³√x + 6

2. Transcendental Function

A. function, which is not algebraic, is called a transcendental function. Trigonometric, Inverse trigonometric, Exponential, Logarithmic, etc are transcendental functions.

Explicit and Implicit Functions

(i) Explicit Functions A function is said to be an explicit function, if it is expressed in the form               y = f(x).

(ii) Implicit Functions A function is said to be an implicit function, if it is expressed in the form            f(x, y) = C, where C is constant.
e.g., sin (x + y) – cos (x + y) = 2

Intervals of a Function

(i) The set of real numbers x, such that a ≤ x ≤ b is called a closed interval and denoted by [a, b] i.e.,     {x: x ∈ R, a ≤ x ≤ b}.

(ii) Set of real number x, such that a < x < b is called open interval and is denoted by (a, b) i.e.,           {x: x ∈ R, a < x < b}

(iii) Intervals [a,b) = {x: x ∈ R, a ≤ x ≤ b} and (a, b] = {x: x ≠ R, a < x ≤ b} are called semiopen and semi-closed intervals.

Graph of Real Functions

1. Constant Function Let c be a fixed real number.

The function that associates to each real number x, this fixed number c is called a constant function i.e., y = f{x) = c for all x ∈ R.

Domain of f{x) = R
Range of f{x) = {c}

2. Identity Function

The function that associates to each real number x for the same number x, is called the identity function. i.e., y = f(x) = x, ∀ x ∈

R. Domain of f(x) = R
Range f(x) = R

3. Linear Function

If a and b be fixed real numbers, then the linear function is defmed as y = f(x) = ax + b, where a and b are constants.

Domain of f(x) = R
Range of f(x) = R

The graph of a linear function is given in the following diagram, which is a straight line with slope a.

4. Quadratic Function

If a, b and c are fixed real numbers, then the quadratic function is expressed as y = f(x) = ax² + bx + c, a ≠ 0 ⇒ y = a (x + b / 2a)² + 4ac – b² / 4a

which is equation of a parabola in downward, if a < 0 and upward, if a > 0 and vertex at ( – b / 2a, 4ac – b² / 4a).

Domain of f(x) = R

Range of f(x) is [ – ∞, 4ac – b² / 4a], if a < 0 and [4ac – b² / 4a, ∞], if a > 0 5. Square Root Function Square root function is defined by y = F(x) = √x, x ≥ 0.

5. Square Root Function

Square root function is defined by y = F(x) = √x, x ≥ 0.

Domain of f(x) = [0, ∞)
Range of f(x) = [0, ∞) 

6. Exponential Function

Exponential function is given by y = f(x) = a^x, where a > 0, a ≠ 1.

7. Logarithmic Function

A logarithmic function may be given by y = f(x) = loga x, where a > 0, a ≠ 1 and x > 0.

The graph of the function is as shown below. which is increasing, if a > 1 and decreasing, if 0 < a < 1.

Domain of f(x) = (0, ∞)
Range of f(x) = R

8. Power Function

The power function is given by y = f(x) = xⁿ ,n ∈ I,n≠ 1, 0. The domain and range of the graph y = f(x), is depend on n.

(a) If n is positive even integer.

i.e., f(x) = x², x⁴ ,….

Domain of f(x) = R
Range of f(x) = [0, ∞)

(b) If n is positive odd integer.

i.e., f(x) = x³, x⁵ ,….

Domain of f(x) = R
Range of f(x) = R

(c) If n is negative even integer.

i.e., f(x) = x^{- 2}, x ^{– 4} ,….

Domain of f(x) = R – {0}
Range of f(x) = (0, ∞)

(d) If n is negative odd integer.

i.e., f(x) = x^{- 1}, x ^{– 3} ,….

Domain of f(x) = R – {0}
Range of f(x) = R – {0}

9. Modulus Function (Absolute Value Function)

Modulus function is given by y = f(x) = |x| , where |x| denotes the absolute value of x, that is

|x| = {x, if x ≥ 0, – x, if x < 0

Domain of f(x) = R
Range of f(x) = [0, &infi;)

Domain of f(x) = R
Range of f(x) = {-1, 0, 1}

Properties of Greatest Integer Function

(i) [x + n] = n + [x], n ∈ I
(ii) x = [x] + {x}, {x} denotes the fractional part of x.
(iii) [- x] = – [x], -x ∈ I
(iv) [- x] = – [x] – 1, x ∈ I
(v) [x] ≥ n ⇒ x ≥ n,n ∈ I
(vi) [x] > n ⇒ x ⇒ n+1, n ∈ I
(vii) [x] ≤ n ⇒ x < n + 1, n ∈ I
(viii) [x] < n ⇒ x < n, n ∈ I
(ix) [x + y] = [x] + [y + x – [x}] for all x, y ∈ R
(x) [x + y] ≥ [x] + [y]
(xi) [x] + [x + 1 / n] + [x + 2 / n] +…+ [x + n – 1 / n] = [nx], n ∈ N

10. Least Integer Function

The least integer function which is greater than or equal to x and it is denoted by (x). Thus, (3.578) = 4, (0.87) = 1, (4) = 4, (- 8.239) = – 8, (- 0.7) = 0

In general, if n is an integer and x is any real number between n and (n + 1).
i.e., n < x ≤ n + 1, then (x) = n + 1

∴ f(x) = (x)

Domain of f = R
Range of f= [x] + 1

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