Lesson 1.1 - Functions: Basic Properties

A function f(x) is defined as a set of all ordered pairs (x, y), such that for each element x, there corresponds exactly one element y.

Figure 1.1-A

The domain of f(x) is the set x. In Figure 1.1-A, the domain is the set of numbers x that have a corresponding value for f(x). Therefore, the domain in this case is x: (-∞,∞)

The range of f(x) is the set y. In Figure 1.1-A, the range is the set of numbers y that have a value. Therefore, the range in this case is y: [0,∞)

How do you know when a graph is not a function?

Functions may not have more than one value of f(x) for an element x. This is why a circle or ellipse is not a function. When these equations are graphed by a calculator, or computer, only the top half is shown. If the bottom was shown, it would not be a proper function because it would fail the vertical line test.

Figure 1.1-B

The circle in Figure 1.1-B is proved to be not a function by applying the vertical line test. Draw a vertical line on a graph, and it is not a function if it crosses more than one point on that graph.

May Functions be combined?

Combinations of functions occur frequently in Calculus. This can be extremely useful whether you’re a Civil Engineer adding up multiple forces in one dimension or a financial analyst taking the present value of an asset that

depends on the stock marker.

Here are the basic combinations of functions in algebraic terms…

If and

Sum:

Difference:

Product:

Quotient:

Composite:

Often students become confused due to wide variety of notation. Here is the other common notation of function combinations: