1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Commonly Occurring Functions The Greatest Integer Function The greatest integer function, f(x) =, is defined as follows: the greatest integer of x is the largest integer less than or equal to x. For instance,= 4 and = -4. Because the greatest integer function consists of only integers, the graph will have discrete jumps or breaks.

2. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Commonly Occurring Functions The Greatest Integer Function: The graph above uses open circles to indicate that the function is either undefined at that point or is defined to be another value. The closed circles are used to emphasize that a certain point really does lie on the graph of the function.

3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Commonly Occurring Functions Piecewise-Defined Function A piecewise-defined function is a function defined in terms of two or more formulas, each valid for its own unique portion of the real number line. In evaluating a piecewise-defined function f at a certain value for x, it is important to correctly identify which formula is valid for that particular value.

4. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The function f is a linear function on the interval and a quadratic function on the interval To graph f, we graph each portion separately, making sure that each formula is applied only on the appropriate interval. The complete graph appears to the right, with the points f(–4) = 6 and f(2) = 4 noted in particular. Also note the use of a closed circle at (–1,0) to emphasize that this point is part of the graph, and the use of an open circle at (–1,1) to indicate that this point is not part of the graph. Example: Commonly Occurring Functions Sketch the graph of the function