1. Chapter 9 Efficiency of Algorithms

2. 9.1 Real Valued Functions

3. Real-Valued Functions of a Real Variable Definition – Let f be a real-valued function of a real variable. 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). – y = f(x) ⇔ the point (x, y) lies on the graph of f.

4. Example

5. Power Functions Definition – Let a be any nonnegative number. Define p a, the power function with exponent a, as follows: p a (x) = x a for each nonnegative real number x.

6. Example p 1/2 = x 1/2 point(x,x 1/2 ) p 0 = x 0 = 1 point(x, 1) p 2 = x 2 point(x, x 2 )

7. Graphing Function on Integers A real-valued function may be graphed on a set of integers.

8. Multiple of a Function Definition – 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, is the real-valued function with the same domain as f that is defined by the rule (Mf)(x) = M* ((f(x)) for all x in the domain of f

9. Example

10. Increasing & Decreasing Functions y is decreasing y is increasing

11. Increasing & Decreasing Function Definition – 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 x 1 and x 2 in S, if x 1 f(x 2 ).

12. Example