1. Fall 2014 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1 Chapter 11.1 Real-Valued Functions of a Real Variable and Their Graphs

2. Cartesian Plane A Cartesian plane or two-dimensional Cartesian coordinate system is a pictorial representation of obtained by setting up a one-to-one correspondence between ordered pairs of real numbers and points in a Euclidean plane. 2 Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Horizontal axis Vertical axis Origin: the intersection of the two axes

3. A Graph of A Function A real-valued function of a real variable is a function from one set of real numbers to another. If f is such a function, then for each real number x in the domain of f, there is a unique corresponding real number f ( x ). The graph of f is the set of all points ( x, y ) in the Cartesian coordinate plane with the property that x is in the domain of f and y = f ( x ). 3 Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

4. A Graph of A Function – cont’ 4 Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University y = f ( x ) the point ( x, y ) lies on the graph of f. Graph of f

5. Power Functions Let a be any nonnegative real number. Define, the power function with exponent a, as follows: for each nonnegative real number x. 5 Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

6. Floor Functions The floor of a number is the integer immediately to its left on the number line. More formally, the floor function F is defined by the rule 6 Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University = the greatest integer that is less than or equal to x = the unique integer n such that

7. Graphing Functions Defined on Sets of Integers Many real-valued functions used in computer science are defined on sets of integers and not on intervals of real numbers. 7 Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

8. Graph of Multiple of a Function Let f be a real-valued function of a real variable and let M be any real number. The function Mf, called the multiple of f by M or M times f, is the real-valued function with the same domain as f that is defined by the rule 8 Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University for all x in domain of f.

9. Increasing and Decreasing Functions Consider the absolute value function, A, which is defined as follows: 9 Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University for all real numbers x.

10. Increasing and Decreasing Functions – cont’ Let f be a real-valued function defined on a set of real numbers, and suppose the domain of f contains a set S. We say that f is increasing on the set S if, and only if, for all real numbers and in S, if then We say that if is decreasing on the set S if, and only if, for all real numbers and in S, if then We say that f is an increasing (or decreasing) function if, and only if, f is increasing (or decreasing) on its entire domain. 10 Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University