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