real-valued functions of a real variable and their graphs lecture 38 section 9.1 mon, mar 28, 2005
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Real-Valued Functions of a Real Variable and Their Graphs
Lecture 38
Section 9.1
Mon, Mar 28, 2005
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
(log10 x)/(log10 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
Growth of the Exponential Function
The exponential functions grow faster and faster as x gets larger and larger.
The Exponential Function f(x) = 2x
1 2 3 4
20
40
60
80
2x
3x4x
Growth of the Exponential Function
The higher the base, the faster the function growsax grows faster then bx, if a > b.
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.