1. Real-Valued Functions of a Real Variable and Their Graphs Lecture 38 Section 9.1 Mon, Mar 28, 2005

2. Functions We will consider real-valued functions that are of interest in studying the efficiency of algorithms. Power functions Logarithmic functions Exponential functions

3. Power Functions A power function is a function of the form f(x) = x a for some real number a. We are interested in power functions where a  0.

4. The Constant Function f(x) = 1

5. The Linear Function f(x) = x

6. The Quadratic Function f(x) = x 2

7. The Cubic Function f(x) = x 3

8. Power Functions x a, a  1 The higher the power of x, the faster the function grows. x a grows faster than x b if a > b.

9. The Square-Root Function

10. The Cube-Root Function

11. The Fourth-Root Function

12. Power Functions x a, 0 < a < 1 The lower the power of x (i.e., the higher the root), the more slowly the function grows. x a grows more slowly than x b if a < b. This is the same rule as before, stated in the inverse.

13. Power Functions x3x3 x2x2 x xx

14. Multiples of Functions x2x2 x 2x 3x

15. Multiples of Functions Notice that x 2 eventually exceeds any constant multiple of x. Generally, if f(x) grows faster than cg(x), for any real number c, then f(x) grows “significantly” faster than g(x). In other words, we think of g(x) and cg(x) as growing at “about the same rate.”

16. Logarithmic Functions A logarithmic function is a function of the form f(x) = log b x where b > 1. The function log b x may be computed as (log 10 x)/(log 10 b).

17. The Logarithmic Function f(x) = log 2 x

18. Growth of the Logarithmic Function The logarithmic functions grow more and more slowly as x gets larger and larger.

19. f(x) = log 2 x vs. g(x) = x 1/n log 2 x x 1/2 x 1/3

20. Logarithmic Functions vs. Power Functions The logarithmic functions grow more slowly than any power function x a, 0 < a < 1.

21. f(x) = x vs. g(x) = x log 2 x x x log 2 x

22. f(x) vs. f(x) log 2 x The growth rate of log x is between the growth rates of 1 and x. Therefore, the growth rate of f(x) log x is between the growth rates of f(x) and x f(x).

23. f(x) vs. f(x) log 2 x x2x2 x 2 log 2 x x log 2 x x

24. Multiplication of Functions If f(x) grows faster than g(x), then f(x)h(x) grows faster than g(x)h(x), for all positive- valued functions h(x). If f(x) grows faster than g(x), and g(x) grows faster than h(x), then f(x) grows faster than h(x).

25. Exponential Functions An exponential function is a function of the form f(x) = a x, where a > 0. We are interested in power functions where a  1.

26. The Exponential Function f(x) = 2 x

27. Growth of the Exponential Function The exponential functions grow faster and faster as x gets larger and larger.

28. The Exponential Function f(x) = 2 x 2x2x 3x3x 4x4x

29. Growth of the Exponential Function The higher the base, the faster the function grows a x grows faster then b x, if a > b.

30. f(x) = 2 x vs. Power Functions (Small Values of x) 2x2x

31. f(x) = 2 x vs. Power Functions (Large Values of x) 2x2x x3x3

32. Growth of the Exponential Function Every exponential function grows faster than every power function. a x grows faster than x b, for all a > 1, b > 0.