Real-Valued Functions of a Real Variable and Their Graphs

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) = xa 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) = x2

7. The Cubic Function f(x) = x3

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

9. The Square-Root Function

10. The Cube-Root Function

11. The Fourth-Root Function

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

13. x3 x2 x x Power Functions

14. 2x 3x x x2 Multiples of Functions

15. Multiples of Functions • Notice that x2 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) = logbx where b > 1. • The function logbx may be computed as (log10x)/(log10b).

17. The Logarithmic Function f(x) = log2x

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

19. x1/2 log2 x x1/3 f(x) = log2x vs. g(x) = x1/n

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

21. x x log2 x f(x) = x vs. g(x) = x log2x

22. f(x) vs. f(x) log2x • 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 xf(x).

23. x2 x2 log2 x x x log2 x f(x) vs. f(x) log2x

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) = ax, where a > 0. • We are interested in power functions where a 1.

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

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

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

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

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

31. 2x x3 f(x) = 2x vs. Power Functions (Large Values of x)

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