describe the relationship between two variables,
acknowledging that one of the variables depends on the other. The wages
of workers who are paid hourly are a function of the number of hours
they work; the quantity of a commodity that consumers are willing to
buy depends on the price the retailer sets for the commodity; the cost
of producing photocopiers depends on the number of photocopiers made.
Regardless of what the variables represent, functions can be described
with words, tables and charts, graphs, and formulas.
Functions are rules that determine ordered pairs of numbers. The values of a function are written (x, y). The value of the x variable determines the value of the y variable; therefore, x is called the independent variable.
Conversely, since y depends on the value of x, y is called the dependent variable. For each x value, there is a unique y value. When graphed on a two-dimensional set of axes, for each x value there should be only one point
corresponding to the unique y
value. The "vertical line test" says that if a vertical line passing
through a graph intersects at more than one point, that graph does not
represent a function. However, if a vertical line passing through a
given graph intersects only once, that graph represents a function.
To express a functional relationship, you write
f(x) = y. Literally, this says
"f of x is equal to y." Or you could say "y is a function of x." Essentially, f is the name of the function that depends on x. (Note that the notation f(x) is simply the name of the function; it does not mean "multiply f by x.") The
letter f is most commonly used as the name of a function.
However, other letters, such as g and h are also used. These functions are written g(x) and h(x).
The domain of a function is the set of all possible values of the independent variable, commonly shown as x. The range of a function consists of all the values of the dependent variable, commonly shown as y or f(x), for which the
function is defined. When functions are graphed, the independent variable is placed on the horizontal axis, or x-axis. The dependent variable is placed on the vertical axis, or y-axis. Because f(x) = y, the y-axis will be labeled f(x).
Consider the following example. The weekly utility costs of a
manufacturing facility depend on the number of days the facility is in
operation. Notice in this example that the facility may operate from
one to five days per week. The table of the utility costs follows.
The cost of utilities (y) is a function of the number of days the facility is operational (x); therefore, y = f(x).
In other words, the cost is determined by the operating days. To
display the function graphically, plot the pairs shown in the table.
In this function, the domain is 1 x 5. The range is 3,000 f(x) 11,000.
This functional relationship can be further described by the formula: y = f(x) = 2,000x +
1,000, where 1 x 5. You can verify this by first substituting the values for x to determine the values
of y, and then comparing these ordered pairs with the numbers in the table.
y = f(x) = 2,000x +
f(1) = 2,000(1) + 1,000 = $3,000
f(2) = 2,000(2) + 1,000 = $5,000
f(3) = 2,000(3) + 1,000 = $7,000
f(4) = 2,000(4) + 1,000 = $9,000
f(5) = 2,000(5) + 1,000 = $11,000
These ordered pairs would be represented as (x, y).
Taking the first example, where x = 1, the ordered pair would be (1, $3,000). The ordered pair for the second example would be (2, $5,000), and so on.
1. Which of the following graphs are functions?
2. Given the following information, what is the corresponding graph?
- f(x) is a function.
- The domain of f(x) is x 0.
- The range of f(x) is y 1.
- The independent variable is the number of units supplied.
- The dependent variable is the price per unit.
3. An employee works from three to eight hours and is paid $15 per
hour. For this employee, the amount earned is a function of the hours
worked. The following questions relate to this function.
- What are the independent and dependent variables?
- What is the domain of the relationship?
- Construct a table of the amount earned for one day, taking into consideration only the correct domain.
- What is the range of the relationship?
4. Use the graph of a demand function below to answer the following questions.
a. What is the quantity demanded at the price of $25?
b. What is the domain of the function?
c. What is the range of the function?
| Why does a marketing brand manager use functions?