382
Relations, Functions, and Matrices
Of course, we also use mathematical functions in algebra and calculus. The
equation g(x) = x
3
expresses a functional relationship between a value for x and
the corresponding value that results when the value for x is used in the equation.
Thus an x value of 2 has the number 2
3
= 8 associated with it. (This number
is expressed as g(2) = 8.) Similarly, g(1) = 1
3
= 1, g(−1) = (−1)
3
= −1, and
so on. For each x value, the corresponding g(x) value is unique. If we were to
graph this function on a rectangular coordinate system, the points (2, 8), (1, 1), and
(−1, −1) would be points on the graph. If we allow x to take on any real number
value, the resulting graph is the continuous curve shown in Figure 5.12.
(–1, –1)
(1, 1)
(2, 8)
g(x)
x
The function in the salary increase example could be described as follows. We set
the stage by the diagram in Figure 5.13, which indicates that the function always
starts with a given degree program and that a particular salary increase is associated with that degree program. The association itself is described by the set of ordered pairs 5(engineering, 2. 25%), (physical sciences, 1. 5%), (computer science,
2. 75%), (liberal arts, 1. 5%), (business, 2. 0%)6.
Computer science
Degree
%
2.75%
Real numbers
Real numbers
g(x)
g
x
For the algebraic example g(x) = x
3
, Figure 5.14 shows that the function
always starts with a given real number and associates a second real number with it.
Figure 5.12
Figure 5.13
Figure 5.14
Degree programs
Salary increases by percent