Chapter 3: Fun with Functions
A function is a rule that connects two sets, the domain and range, by assigning to each element in the domain one and only one element in the range.
A function can be specified by means of a table that lists each element of the domain and the corresponding element of the range. A function may also be specified by its rule. Mathematical functions can be specified by their formulae.
It is traditional to use x to represent elements in the domain and y for elements in the range and to write y=f(x) to represent the function, f.
Problems: suppose the domain is the set of children and the range is the set of men. (1) Can a function be constructed to assign a biological father to each child? (2) Can a function be defined using men as the domain and children as the range assigning children to fathers? (Answer: No. Because each man may have more than one or even zero children.) (3) Can a function be defined on the domain of men to the range of integers that assigns the number of children to each man? (Answer: Yes).
y = f(x)
The composition of two functions, y=f(x), and z=g(y) is the function, z=g(f(x)), that assigns a value of z to each element of x.
Inverse of a function: some functions have an inverse function. Some do not. If it exists, the inverse is the function that assigns x to y. If y = f(x), and if f^{–1} is the inverse, then
x = f^{–1}(y)
x = f^{–1}(f(x))
Graph of a function: It is customary to plot y on the ordinate and x on the abscissa. Hint: First, construct a table, then draw the graph.
Monotonic Function
A strictly increasing monotonic function , f, has the property that
x_{2} > x_{1} if and only if f(x_{2}) > f(x_{1})
A weakly increasing monotonic function, f, has the property that
x_{2} > x_{1} if and only if f(x_{2}) > f(x_{1}) (note the > instead of >)
Linear function:
y = ax+b
a is the slope = (change in y)/(change in x).
b is the intercept (when x = 0, y = b)
The inverse of a linear function is linear.
Ex. Find the inverse of
y = 2x + 2
The composition of two linear functions is linear.
Ex. If y = 2x + 2 and z = 3y + 1,
Find the composition, expressing z as a function of x.
The graph of a linear function is a straight line.
Ex. Draw a graph of these 3 linear functions:
y = .5x
y = x
y = 2x
for x = –5 to 5
Ex. Draw a graph of these linear functions:
y = x – 2
y = x
y = x + 3
Ex. Draw a graph of these three functions:
y = 2x
y = x + 3
y = 3x – 3
Where do these 3 functions intersect?
Accelerated Functions
Linear functions have a constant slope. In other words, the slope (increase in y divided by increase
in x) is the same everywhere. The acceleration is zero.
Accelerated functions have a slope that changes as a function of the value of x.
Positively accelerated function: the slope increases as x increases. For example, .
Negatively accelerated function: the slope decreases as x increases. For example, y = .
Power Functions
a = multiplicative coefficient
b = exponent ("power")
The inverse of a power function is a power function with the reciprocal exponent.
Ex. Find the inverse of
The composition of two power functions is a power function with exponent equal the product of the exponents.
and
Ex. Find the composition of
and
(Express z as a function of x.)
Ex. Draw a graph showing these four functions, from 0 to 2, in steps of .2.
y = x
Ex. Draw a graph showing these four functions:
Exponential and Logarithmic Functions
The exponential is the function whose slope is proportional to its height. Therefore, exponential functions are sometimes called "growth" functions, because they can be used to represent a growing total (population or investment) that grows at a constant proportion of its value. The equation of an exponential function is as follows:
y = ab^{x}
a = coefficient
b = base
The exponential is sometimes written: y = exp(x)
Do not confuse power function with exponential!
The logarthmic function is the inverse of the exponential. The logarithm of x is also the integral of the equation,
f(t) = bdt/t, integrated from t = 1 to x.
When b = 1, we have the natural logarithim, or log to the base e,
where e is approximately 2.718.
Log(1) = 0 (This is true for logs with all bases)
Log of a product is the sum of the logs:
log(xy) = log(x) + log(y)
Log of a ratio is the difference of the logs:
log(x/y) = log(x) – log(y)
log(1/x) = log(1) – log(x) = –log(x)
Log of a power function is a linear function:
log (ax^{b}) = b log(x) + log(a)
(This property can be used to fit power functions to data).(These properties also imply that log functions with different bases are proportional to one another).
The exponential of a sum is the product of exponentials
exp(x + y) = exp(x) exp(y)
The exponential of a difference is a ratio of exponentials
exp(x – y) = exp(x)/exp(y)
The exponential of a product is a power function of the exponential:
exp(xy) = [exp(x)]^{y} = [exp(y)]^{x}
(This property implies that exponential functions with different bases are power functions of one another. The composition of an exponential function and its corresponding log function is the identity function; the composition of exponential and a linear function of the log is a power function).
Table of Common Functions
x 
Log_{10}(x) 
10^{x} 
Lnx 
Exp(x) 
x^{ 0.5} 
x^{ 2} 
0 
undef 
1 
undef 
1.000 
0.000 
0 
1 
0.000 
10 
0.000 
2.718 
1.000 
1 
2 
.301 
100 
.639 
7.389 
1.414 
4 
3 
.477 
1000 
1.099 
20.086 
1.732 
9 
4 
.602 
10000 
1.386 
54.598 
2.000 
16 
5 
.699 
100000 
1.609 
148.413 
2.236 
25 
6 
.778 
etc 
1.792 
403.428 
2.449 
36 
7 
.845 

1.946 
1096.63 
2.646 
49 
8 
.903 

2.079 
2981.0 
2.828 
64 
9 
.954 

2.197 
etc 
3.000 
81 
10 
1.000 

2.303 

3.162 
100 
100 
2.000 

4.605 

10.000 
10^{4} 
1000 
3.000 

6.908 

31.623 
10^{6} 
A. Plot y = logx over the domain from x = .1 to 10
B. Plot y = x.5 over the domain from 0 to 10.
C. Plot y = x2 over the domain from x = 0 to 10.
D. Plot y = Exp(x) over the domain from x = 0 to 2 in steps of .2.
*E. Write BASIC programs to calculate the values of these functions; check this table for errors. Hint: Here is a simple program to try:
10 FOR X = 1 TO 10 STEP .5
20 PRINT "X="; X, "LOG="; LOG(X)
30 NEXT
F. Plot log X vs. lnX. What does the slope represent?
(i.e., plot log10x vs. (natural log) = logex = lnx)
For any bases, b and c, logb(x) = logc(x)/logc(b)
G. Learn how to access these functions on your scientific calculator and on your computer. Learn how to access these functions from Excel.
by Michael H. Birnbaum, © 19742001, all rights reserved.