Let A and B be .two non-empty sets, then a function f from set A to set B is a rule which associates 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, a_{0}, a_{1}, a_{2},…, a_{n} are real numbers and n is any non -negative integer, then f (x) is called a polynomial function in x.

If a_{0} ≠ 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) = 3x^{5} – 4x^{2} – 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 non-integral 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 semi-open 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) = ax^{2} + bx + c, a ≠ 0

⇒ y = a (x + b / 2a)^{2} + 4ac – b^{2} / 4a

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

Domain of f(x) = R

Range of f(x) is [ – ∞, 4ac – b^{2} / 4a], if a < 0 and [4ac – b^{2} / 4a, ∞], if a > 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.

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

Domain of f(x) = R

Range of f(x) = (0, ∞)

7. **Logarithmic Function**

A logarithmic function may be given by y = f(x) = log_{a} 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) = x^{n} ,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) = x^{2}, x^{4} ,….

Domain of f(x) = R

Range of f(x) = [0, ∞)

(b) If n is positive odd integer.

i.e., f(x) = x^{3}, x^{5} ,….

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}

11. **Greatest Integer Function**

The greatest integer function is defined as

y = f(x) = [x]

where, [x] represents the greatest integer less than or equal to x. i.e., for any integer n, [x] = n, if n ≤ x < n + 1 Domain of f(x) = R Range of f(x) = I

**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

12. **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

13. **Fractional Part Function**

It is denoted as f(x) = {x} and defined as

(i) {x} = f, if x = n + f, where n ∈ I and 0 ≤ f < 1

(ii) {x} = x – [x]

i.e., {O.7} = 0.7, {3} = 0, { – 3.6} = 0.4

(iii) {x} = x, if 0 ≤ x ≤ 1

(iv) {x} = 0, if x ∈ I

(v) { – x} = 1 – {x}, if x ≠ I

**Graph of Trigonometric Functions**

1. **Graph of sin x**

(i) Domain = R

(ii) Range = [-1,1]

(iii) Period = 2π

2. **Graph of cos x**

(i) Domain = R

(ii) Range = [-1,1]

(iii) Period = 2π

3. **Graph of tan x**

(i) Domain = R ~ (2n + 1) π / 2, n ∈ I

(ii) Range = [- &infi;, &infi;]

(iii) Period = π

4. **Graph of cot x**

(i) Domain = R ~ nπ, n ∈ I

(ii) Range = [- &infi;, &infi;]

(iii) Period = π

5. **Graph of sec x**

(i) Domain = R ~ (2n + 1) π / 2, n ∈ I

(ii) Range = [- &infi;, 1] ∪ [1, &infi;)

(iii) Period = 2π

6. **Graph of cosec x**

(i) Domain = R ~ nπ, n ∈ I

(ii) Range = [- &infi;, – 1] ∪ [1, &infi;)

(iii) Period = 2π

**Operations on Real Functions**

Let f: x → R and g : X → R be two real functions, then

(i) **Sum** The sum of the functions f and g is defined as

f + g : X → R such that (f + g) (x) = f(x) + g(x).

(ii) **Product** The product of the functions f and g is defined as

fg : X → R, such that (fg) (x) = f(x) g(x)

Clearly, f + g and fg are defined only, if f and g have the same domain. In case, the domain of f and g are different. Then, Domain of f + g or fg = Domain of f ∩ Domain of g.

(iii) **Multiplication by a Number** Let f : X → R be a function and let e be a real number .

Then, we define cf: X → R, such that (cf) (x) = cf (x), ∀ x ∈ X.

(iv) **Composition** (Function of Function) Let f : A → B and g : B → C be two functions.

We define gof : A → C, such that

got (c) = g(f(x)), ∀ x ∈ A

**Alternate** There exists Y ∈ B, such that

if f(x) = y and g(y) = z, then got (x) = z

**Periodic Functions**

A function f(x) is said to be a periodic function of x, provided there exists a real number T > 0, such that

F(T + x) = f(x), ∀ x ∈ R

The smallest positive real number T, satisfying the above condition is known as the period or the fundamental period of f(x) ..

**Testing the Periodicity of a Function**

(i) Put f(T + x) = f(x) and solve this equation to find the positive values of T independent of x.

(ii) If no positive value of T independent of x is obtained, then f(x) is a non-periodic function.

(iii) If positive val~es ofT independent of x are obtained, then f(x) is a periodic function and the least positive value of T is the period of the function f(x).

**Important Points to be Remembered**

(i) Constant function is periodic with no fundamental period.

(ii) If f(x) is periodic with period T, then 1 / f(x) and. √f(x) are also periodic with f(x) same period T.

{iii} If f(x) is periodic with period T_{1} and g(x) is periodic with period T_{2}, then f(x) + g(x) is periodic with period equal to LCM of T_{1} and T_{2}, provided there is no positive k, such that f(k + x) = g(x) and g(k + x) = f(x).

(iv) If f(x) is periodic with period T, then kf (ax + b) is periodic with period T / |a|’ where a, b ,k ∈ R and a, k ≠ 0.

(v) sin x, cos x, sec x and cosec x are periodic functions with period 2π.

(vi) tan x and cot x are periodic functions with period π.

(vii) |sin x|, |cos x|, |tan x|, |cot x|, |sec x| and |cosec x| are periodic functions with period π.

(viii) sin^{n} x, cos^{n} x, sec^{n} x and cosec^{n}x are periodic functions with period 2π when n is odd, or π when n is even .

(ix) tan^{n} x and cot^{n} x are periodic functions with period π.

(x) |sin x| + |cos x|, |tan x| + |cot x| and |sec x| + |cosec x| are periodic with period π / 2.

(xi) If f(x) is a periodic function with period T and g(x) is any function, such that domain of f ⊂ domain of g, then gof is also periodic with period T.

**Even and Odd Functions**

**Even Functions** A real function f(x) is an even function, if f( -x) = f(x).

**Odd Functions** A real function f(x) is an odd function, if f( -x) = – f(x).

**Properties of Even and Odd Functions**

(i) Even function ± Even function = Even function.

(ii) Odd function ± Odd function = Odd function.

(iii) Even function * Odd function = Odd function.

(iv) Even function * Even function = Even function.

(v) Odd function * Odd function = Even function.

(vi) gof or fog is even, if anyone of f and g or both are even.

(vii) gof or fog is odd, if both of f and g are odd.

(viii) If f(x) is an even function, then d / dx f(x) or ∫ f(x) dx is odd and if dx .. f(x) is an odd function, then d / dx f(x) or ∫ f(x) dx is even.

(ix) The graph of an even function is symmetrical about Y-axis.

(x) The graph of an odd function is symmetrical about origin or symmetrical in opposite quadrants.

(xi) An even function can never be one-one, however an odd function mayor may not be one-one.

**Different Types of Functions** (Mappings)

1. **One-One and Many-One Function**

The mapping f: A → B is a called one-one function, if different elements in A have different images in B. Such a mapping is known as injective function or an injection.

**Methods to Test One-One**

(i) **Analytically** If x_{1}, x_{2} ∈ A,

then f(x_{1}) = f(x_{2}) => x_{1} = x_{2}

or equivalently x_{1} ≠ x_{2} => f(x_{1}) ≠ f(x_{2})

(ii) **Graphically** If any .line parallel to x-axis cuts the graph of the function atmost at one point, then the function is one-one.

(iii) **Monotonically** Any function, which is entirely increasing or decreasing in whole domain, then f(x) is one-one.

**Number of One-One Functions** Let f : A → B be a function, such that A and B are finite sets having m and n elements respectively, (where, n > m).

The number of one-one functions

n(n – 1)(n – 2) …(n – m + 1) = { ^{n}P_{m}, n ≥ m, 0, n < m

The function f : A → B is called many – one function, if two or more than two different elements in A have the same image in B.

2. **Onto** (Surjective) **and Into Function**

If the function f: A → B is such that each element in B (codomain) is the image of atleast one element of A, then we say that f is a function of A ‘onto’ B. Thus, f: A → B, such that f(A) = B.

i.e., Range = Codomain

Note Every polynomial function f: R → R of degree odd is onto.

**Number of Onto** (surjective) **Functions** Let A and B are finite sets having m and n elements respectively, such that 1 ≤ n ≤ m, then number of onto (surjective) functions from A to B is

^{n}Σ_{r = 1} (- 1)^{n – r}^{n}C_{r} r^{m}

= Coefficient of x^{n} in n! (e^{x} – 1)^{r}

If f : A → B is such that there exists atleast one element in codomain which is not the image of any element in domain, then f(x) is into.

Thus, f : A → B, such that f(A) ⊂ B

i.e., Range ⊂ Codomain

**Important Points to be Remembered**

(i) If f and g are injective, then fog and gof are injective.

(ii) If f and g are surjective, then fog is surjective.

(iii) Iff and g are bijective, then fog is bijective.

**Inverse of a Function**

Let f : A → B is a bijective function, i.e., it is one-one and onto function.

We define g : B → A, such that f(x) = y => g(y) = x, g is called inverse of f and vice-versa. Symbolically, we write g = f^{-1}

Thus, f(x) = y => f^{-1}(y) = x

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