Standard Real Functions and their Graphs

Constant Function: If is a fixed real number, then a function given by for all is called a constant function. Sometimes we also call it the constant function .

Identity Function: The function that associates each, real number to itself is called the identity function and is usually denoted by . Thus, the function defined by for all is called the identity function.

Clearly, the domain and range of the identity function are both equal to . The graph of the identity function is a straight line passing through the origin and inclined at an angle of with X-axis.

Modulus Function: The function defined by is called modulus function.

It is also called the absolute value function. We observe that the domain of the modulus function is the set of all real numbers and the range is the set of all non-negative real numbers i.e.

*Properties of Modulus Function:*

i) For any real number

ii) If are positive real numbers, then

iii) For real numbers x and y, we have

Greatest Integer Function: For any real number & we use the symbol or, to denote the greatest integer less than or equal to .

The function defined by for all is called the greatest integer function or the floor function.

It is also called a step function. Clearly, domain of the greatest integer function is the set of all real numbers and the range is the set of all integers as it attains only integer values.

Properties of Greatest Integer Function: If is an integer and is a real number between and , then:

i)

ii) for any integer

iii)

iv)

v)

vi)

vii)

viii)

ix)

x)

xi)

Smallest Integer Function: The function defined by for all is called the smallest integer function or the ceiling function.

It is also a step function. We observe that the domain of the smallest integer function is the set of all real numbers and its range is the set of all integers.

Properties of Smallest Integer Function: Following are some properties of smallest

integer function:

i)

ii)

iii)

iv)

v)

Fractional Part Function: For any real number we use the symbol to denote the fractional part or decimal part of .

The function defined by for all is called the fractional part function. The domain of the fractional part function is the set of all real numbers and the range of the set .

Therefore from the definition, for all

Signum Function: The function defined by:

or

The domain of the signum function is the set of all real numbers and the range is the set of

Exponential Function: If is a positive real number other than unity, then a function that associates each to is called the exponential function.

Or a function defined by , where and is called the exponential function.

The domain of an exponential function is the set of all real numbers and and the range is the set as it attains only positive values.

Case 1: When , the values of increase as the values of increase.

Case 2: When , In this case, the values of decrease with the increase in and for all .

Logarithmic Function: If and , then the function defined by is called logarithmic function.

Note that logarithmic function and the exponential function are inverse functions i.e.

The domain of the logarithmic function is the set of all non-negative real numbers i.e. and the range is the set R of all real numbers.

Case 1: When

The values of increase with the increase in .

Case 2: When

The values of decrease with the increase in .

Properties of Logarithmic Functions:

i) where

ii) where

iii) , where and

iv) , where and

v) , where and

vi) , where and

vii)

viii) If , then the values of increase with the increase in .

Also

ix) If , then the values of decrease with the increase in . i.e

Also

x) for and

*Note:* Functions and are inverse of each other. So, their graphs are mirror images of each other in the line mirror .

Reciprocal Function: The function that associates a real number to its reciprocal is called the reciprocal function. Since is not defined for we define the reciprocal function as follows:

The function defined by is called a reciprocal function.

The domain of the reciprocal function is and its range is also .

The sign of is the same as that of and decreases with the increase in .

Square Root Function: The function that associates a real number to is called the square root function. Since is real for . So, we defined the square root function as follows:

The function defined by is called the square root function.

The domain of the square root function is i.e. and its range is also .

Square Function: The function that associates a real number to its square i.e. is called the square function. Since is defined for all . So, we define the square function as follows:

The function defined by is called the square function. Clearly, domain of the square function is and its range is the set of all non-negative real numbers i.e. .

Cube Function: The function that associate a real number to its cube is called the cube function. If is meaningful for all we define the cube function as follows:

The function defined by is called the cube function.

The sign of and will always be the same. When increases, increase too. The graph is symmetrical in opposite quadrant.

Cube Root Function: The function that associate a real number to its cube root is called the cube root function

The function defined by is called the cube function.

Domain and range of the cube root function are both equal to .