FUNCTIONS

WHEN ONE THING DEPENDS on another, as, for example, the area of a circle depends on the radius, or the temperature on the mountain depends on the height, then we say that the first is a "function" of the other. The area of a circle is a function of -- it depends on -- the radius.
Mathematically:
A relationship between two variables, typically x and y, is called a function, if there is a rule that assigns to each value of x one and only one value of y.
Thus a "function" must be single-valued ("one and only one"). For example,
y = 2x + 3.
To each value of x there is a unique value of y.
The values (Topic 2) that x may assume are called the domain of the function. We say those are the values for which the function is defined.
In the function y = 2x + 3, the domain may include all real numbers (Topic 2). x could be any real number. Or, as in Example 1 below, the domain may be arbitrarily restricted.
There is one case however in which the domain must be restricted: A denominator may not be 0. In this function,
y
=
1 x − 2
,
x may not take the value 2. For, division by 0 is an excluded operation. (Lesson 6 of Algebra.)
Once the domain has been defined, then the values of y that correspond to each value of x, are called the range. Thus if 5 is a value in the domain of y = 2x + 3, then y = 2· 5 + 3 = 13 is the corresponding value in the range.
By the value of the function we mean the value of y. Again, when x = 5, we say that the value of the function is 13. The range, then, is composed of the values of the function.
It is customary to call x the independent variable, because we are given, or we must choose, the value of x first. y is then called the dependent variable, because its value will depend on the value of x.
Example 1. Let the domain of a function be this set of values:
A = {0, 1, 2, −2}
and let the variable x assume each value. Let the rule that relates the value of y to the value of x be the following:
y = x² + 1.
a) Write the set of ordered pairs (x, y) which "represents" this function.
Answer. {(0, 1), (1, 2), (2, 5), (−2, 5)}
That is, when x = 0, then y = 0² + 1 = 1.
When x = 1, then y = 1² + 1 = 2. And so on.
b) Write the set B which is the range of the function.
Answer. B = {1, 2, 5, 5}. The values in the range are simply those values of y that correspond to each value of x.
Notice that to each value of x in the domain there corresponds one -- and only one -- value of the function. Even though the value 5 is repeated, it is still one and only one value.
Example 2. Here is a relationship in which y is not a function of x:
y² = x
When x = 4, for example -- y² = 4 -- then y = 2 or −2. To each value of x, there is more than one value of y.
Problem 1. Let y be a function of x as follows:
y = 3x²