# Floor Function

## Floor Function

Function f from $$\mathbb{R}$$ to $$\mathbb{R}$$ such that for every real number xf(x) is equal to the greatest integer less than or equal to x.

### Notation

• The integer part of x is denoted by [x].
• The relationship defined by (x) = [x] defines the basic model of the floor function.

### Properties

• The general model, which is translated by a step function is defined by the relation f(x) = a[bx] , in which parameter a characterizes a “riser” (or vertical jump) on the graph, and parameter b characterizes the length of a horizontal step or each horizontal segment of the graph.
• Parameter a causes a vertical scale change of value |a| and affects the orientation—or steepness—of the graph depending on whether a is positive or negative.
• Parameter b causes a horizontal scale change of value $$\left| \dfrac {1}{b} \right|$$ and affects the orientation – or steepness – of the graph, depending on whether b is positive or negative.
• The standard form of the general model is$$f(x) = \textrm{a}\left[\textrm{b}\left(x – \textrm{h}\right)\right] + \textrm{k}$$ where parameters h and k characterize a horizontal translation and a vertical translation, respectively, of the graph of the basic relation.

### Example

The graph of the function defined by the rule f[x] is that of a step function.

The following graph represents the function defined by the relation f[x] = −3[0,5x].

In this case, the direction of the graph was reversed, as parameter a is negative. The graph underwent a scale change of 3 units along the vertical axis and a scale change of |2| units along the horizontal axis.