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Science > Mathematics > Functions > You are Here |

#### Real Function:

- A function whose domain and co-domain are the set or subset of real numbers R, then the function is called a real function.

Thus if ƒ: R → R then ƒ is a real function. **Example:**

Consider function y = ƒ(x) = x^{2} + 3x + 2

For every x ∈ R, y = ƒ(x) = x^{2} + 3x + 2 ∈ R,

Thus the function y = ƒ(x) = x^{2} + 3x + 2 is a real function.

#### Constant Function:

- If a real function ƒ is defined as ƒ(x) = k, k is constant for all x ∈ R Then is called a constant function.
**Example:**ƒ(x) = 3, ƒ(x) = – 4 etc.

#### Note:

- The domain for the constant function is a set of real number R, i.e. D
_{ƒ }= R, while its range is {k} - The range contains only one element. i.e. R
_{ƒ}= {k} - Constant function is many-one function
- The graph for a constant function is as follows.

#### Zero Function:

- For constant function k = 0 then the function is called zero function.
**Example:**ƒ(x) = 0, g(x) = 0 etc

#### Note:

- The domain for zero function is a set of real number R i.e. D
_{ƒ }= R, while its range is {0}. - The range contains only one element i.e. zero. i.e. R
_{ƒ }= {0} - Zero function is many-one function.
- The graph for the zero function is as follows.

The graph is x-axis

#### Identity Function:

- If real function ƒ is defined as ƒ(x) = x, for all x ∈ R Then ƒ is called identity function.
**Example:**y = x

#### Note:

- The domain and range for identity function is a set of real number R i.e. D
_{ƒ }= R. - The graph for the zero function is as follows.

#### Absolute value function:

- A function ƒ is defined by ƒ(x) = |x|, Where

is called an absolute value function.

#### Note:

- The domain for absolute value function is a set of real number R i.e. D
_{ƒ }= R. - The graph for the absolute value function is as follows.

#### Signum function:

- A function is defined by

is called signum function.

#### Note:

- The domain for signum function is a set of real number R i.e. D
_{ƒ }= R. - The range of signum function contains three elements only. = {-1, 0, 1}
- The graph for the signum function is as follows.

#### Greatest integer function:

- The greatest integer function ƒ is defined as [x], the greatest integer ≤ x, for each x ∈ R.
- Thus, [x] = x if x is integer and [x] = an integer immediately on the left side of x if x is not an integer.
- Examples:

[5] = 5, [-6.9] = -7, [0] = 0, [2.3] = 2, [17/3] = 5

#### Note:

- The domain for greatest integer function is a set of real number R i.e. D
_{ƒ }= R. - The range of greatest integer function is a set of integers. R
_{ƒ }= I - The graph for the greatest integer function is as follows.

#### Fractional part function:

- A functionƒ defined by ƒ(x) = x – [x], is called fractional part function.
**Examples:**(3.9) =3.9 -3 = 0.9 and (-6.9) = -6.9 – (-7) = 0.1

#### Note:

- The domain for fractional part function is a set of real number R i.e. D
_{ƒ }= R. - Range of fractional part function is R
_{ƒ }= [0, 1) i.e. 0 ≤ f(x) < 0 - The graph for the fractional part function is as follows.

#### Linear function:

- A function defined by ƒ(x) = mx + c, where m, c ∈ R and m ≠ 0 is called a linear function.
**Example:**ƒ(x) = y = 3x + 5

#### Note:

- The domain for a linear function is a set of real number R i.e. D
_{ƒ }= R. - The range of a linear function is a set of real numbers. R
_{ƒ }= R - The graph of a linear function is a straight line.
- If c = 0 then the graph passes through the origin.

#### Polynomial Function:

- If real function ƒ is defined as ƒ(x) = a
_{0}+ a_{1}x + a_{2}x^{2}+ a_{3}x^{3}+ ……… +a_{n}x^{n}. Where a_{0}, a_{1}, a_{2}, a_{3}, …,a_{n}∈ R and n is a whole number. Then ƒ is called as a polynomial function.

Example : ƒ(x) = x^{2}+ 3x + 2

#### Note:

- The domain and range for a polynomial function is a set of real number R. Thus, D
_{ƒ }= R. and R_{ƒ }= = R

#### Reciprocal function:

- A function ƒ defined by ƒ(x) = 1/x, Where x ∈ R and x ≠ 0. is called reciprocal function.

#### Note:

- The domain for reciprocal function is a set of real number R except x ≠ 0 i.e. D
_{ƒ }= R – {0}. - The graph for the reciprocal function is as follows.

#### Exponential function:

- A function ƒ defined by ƒ(x) = e
^{x}is called exponential function.

#### Note:

- The domain for the exponential function is a set of real number R i.e. D
_{ƒ }= R - The graph for the exponential function is as follows.

#### Logarithmic function:

- A function ƒ defined by ƒ(x) =log x, x > 0 is called logarithmic function.

#### Note:

- The domain for the logarithmic function is a set = {x| x ∈ R and x > 0}
- The graph for the logarithmic function is as follows.

#### Trigonometric functions:

**Graphs of Trigonometric Fumctions:**

**Inverse trignometric functions:**

#### Some Important Results of Inverse Functions:

#### SET – I

- sin(sin
^{-1}x) = x, for |x|<1 - sin
^{-1}(sin x) = x, for |x| ≤ π/2 - cos(cos
^{-1}x) = x, for |x|<1 - cos
^{-1}(cos x) = x, for x = [0, π] - tan(tan
^{-1}x) = x, ∀ x ∈ R - tan
^{-1}(tan x) = x, for x ∈ (- π/2, π/2) - cot(cot
^{-1}x) = x, x ∈ R - cot
^{-1}(cot x) = x, for x ∈ (0, π) - sec(sec
^{-1}x) = x, |x| ≥ 1 - sec
^{-1}(sec x) = x, for x ∈ [0, – π/2) ∪ (π/2, π] - cosec(cosec
^{-1}x) = x, |x| ≥ 1 - cosec
^{-1}(cosec x) = x, for x ∈ [- π/2, 0) ∪ (0, π/2]

#### SET- II

- cosec
^{-1}x = sin^{-1}(1/x) - sec
^{-1}x = cos^{-1}(1/x) - cot
^{-1}x = tan^{-1}(1/x)

#### SET – III

1. sin^{-1}x + cos^{-1}x = π/2

2. tan^{-1}x + cot^{-1}x = π/2

3. sec^{-1}x + cosec^{-1}x = π/2

#### SET – IV

#### SET – V

Rational functions:

- A function ƒ of the form p(x)/q(x) = 0, q(x) ≠ 0 is called rational function. Its domain is R except for q(x) ≠ 0

Science > Mathematics > Functions > You are Here |

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