What is least integer function Difference between greatest integer function and least integer function

The greatest integer function of a real number x is represented by f(x) = [x] or |_x_| the greatest integer function returns the largest integer less than or equal to x for all real numbers x. we can say that the greatest integer function rounds down a real number to the nearest integer The domain of the greatest integer function is the set of real numbers which is divided into a number of intervals like [-3, 2), [-2, 1), [-1, 0), [0, 1), [1, 2), and so on [a, b) is just an interval notation which means a ? x < b.. for example : [6.8] =6 , [4] = 4 , [-5.7] = - 6.. Greatest integer function is also called floor function.. GIF is not a periodic function. Though function is defined for all real x, but graph is not continous. It breaks at integral values of x. Domain=R Range=Z We have seen that greatest integer function represents the integer, which can be considered to be the floor integral value of a real number. Correspondingly, we define a ceiling function called “least integer function (LIF)”, which returns the least integer greater than or equal to the number (x). We denote least integer function as “[x)” or "(x)". Some authors reserve "(x)" for near integer function. It is not important as we can always specify what we mean by qualifying the symbol explicitly. We interpret LIF as : [x) = least integer greater than or equal to the number x [x) = least integer not less than or equal to the number x Clearly, least integer function returns a value, which is the integral “ceiling” of the number. For this reason, least integer function is also known as “ceiling” function. Working rules for finding least integer function are : If “x” is an integer, then [x) = x. If “x” is not an integer, then [x) evaluates to least integer greater than “x”. The value of f(x) is an integer (n) such that : f(x)=n;ifn?1read less

Greatest integer function also called as the floor function is one of the generally used function in mathematics. The greatest integer function of a real number x is represented by [ x ] or | x |. This function is also known by the names of step or floor function. It returns the greatest integer less than or equal to x. Another way to understand it that the greatest integer function rounds down the values into the nearest integer possible. This function is piece wise defined. This function basically gives us the nearest integer value of the input as an output.It is represented by the square bracket ( [X] ). Definition of function is given below, f(x) = [X] = greatest integer <= Domain of function: {R}, set of real number Range: { I }, Set of integer value. Examples: [2.8] = 2 (greatest integer less than and equal to 2.8 ) [3.999] = 3 (greatest integer less than and equal to 3.999) [4] = 4 (as 4 is itself an integer, remember the equal to sign) [-5.99] = -6 (the greatest integer less than -5.99 is -6, remember the sign less than or equal to). Smallest integer function also called as Ceil function is one of the generally used function in mathematics. This function basically, gives us the nearest integer value of the input as an output. It is represented by the square bracket ( [X] ). Definition of function is given below, f(x) = [X] = smallest integer Domain of function: {R}, set of real number Range: { I },Set of integer value. Examples: [2.8] = 3 (smallest integer greater than and equal to 2.8 ) [3.999] = 4 (smallest integer greater than and equal to 3.999) [4] = 4 (as 4 is itself an integer, remember the equal to sign) [-5.99] = -5(smallest integer greater than and equal to -5.99 is -5) For any real number x, this step function is defined by, f(x) = |x| where, the smallest integer is greater than or equal to x. For example, [4.75] = 5, [-7.2] = -7, [4] = 4. read less