Real-Valued Functions of a Real Variable and Their Graphs

Lecture 43

Section 9.1

Wed, Apr 18, 2007

Functions

We will consider real-valued functions that are of interest in studying the efficiency of algorithms.Power functionsLogarithmic functionsExponential functions

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.

The Constant Function f(x) = 1

2 4 6 8 10

0.5

1

1.5

2

The Linear Function f(x) = x

2 4 6 8 10

2

4

6

8

10

The Quadratic Function f(x) = x2

2 4 6 8 10

20

40

60

80

100

The Cubic Function f(x) = x3

2 4 6 8 10

100

200

300

400

500

600

Power Functions xa, a 1

The higher the power of x, the faster the function grows.xa grows faster than xb if a > b.

The Square-Root Function

2 4 6 8 10

0.5

1

1.5

2

2.5

3

The Cube-Root Function

2 4 6 8 10

0.5

1

1.5

2

The Fourth-Root Function

2 4 6 8 10

0.25

0.5

0.75

1

1.25

1.5

1.75

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.

0.5 1 1.5 2

1

2

3

4

Power Functions

x3x2

x

x

Multiples of Functions

1 2 3 4

2.5

5

7.5

10

12.5

15 x2

x

2x

3x

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.”

Logarithmic Functions

A logarithmic function is a function of the form

f(x) = logb x

where b > 1. The function logb x may be computed as

(ln x)/(ln b).

The Logarithmic Function f(x) = log2 x

10 20 30 40 50 60

-2

2

4

6

Growth of the Logarithmic Function

The logarithmic functions grow more and more slowly as x gets larger and larger.

f(x) = log2 x vs. g(x) = x1/n

5 10 15 20 25 30

-2

2

4 log2 x

x1/2

x1/3

Logarithmic Functions vs. Power Functions

The logarithmic functions grow more slowly than any power function xa, 0 < a < 1.

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

0.5 1 1.5 2 2.5 3

1

2

3

4

x

x log2 x

f(x) vs. f(x) log2 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).

2 4 6 8

10

20

30

40

50

f(x) vs. f(x) log2 x

x2x2 log2 x

x log2 x

x

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).

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.

The Exponential Function f(x) = 2x

1 2 3 4

2.5

5

7.5

10

12.5

15

The Exponential Function f(x) = 2x

1 2 3 4

20

40

60

80

2x

3x4x

Growth of the Exponential Function

The larger the base, the faster the function growsax grows faster then bx, if a > b > 1.

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

0.5 1 1.5 2

1

2

3

4

5

2x

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

5 10 15 20

500

1000

1500

2000

2500

3000

3500

2x

x3

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.

Rates of Growth of Functions

The first derivative of a function gives its rate of change, or rate of growth.

Rates of Growth of Power Functions

.10 if ,decreasing is,1 if constant, is

,1 if ,increasing is1

aa

aaxx

dx

d aa

Rates of Growth of Logarithmic Functions

.1 if ,decreasing is ln

1log b

bxx

dx

db

Rates of Growth of Exponential Functions

aaadx

d xx ln