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Unformatted text preview: Section 1.1: Four Ways to Represent a Function 1. The Definition of a Function Functions are one of the most basic tools in mathematics, so we start by considering the definition of a function and all related concepts. Definition 1.1. A function f is a rule which assigns to each element in a set D , called the domain of f , exactly one element, called f ( x ), in a set R . There are a number of important concepts and new terminology related to functions: ( i ) If x is in the domain of f , we call f ( x ) the value of f at x . ( ii ) The range of f is the set of all values that f can take. ( iii ) An element from the domain of f ( x ) is called an input of f ( x ). ( iv ) An element from the range of f ( x ) is called an output of f ( x ). ( v ) A symbol representing an arbitrary element from the domain of f is called an independent variable . We usually use x . ( vi ) A symbol representing range values of f is called a dependent variable (because it depends upon the independent variable). We usually use y . A couple of important and often overlooked facts about functions are the following: ( i ) The range of a function depends upon the domain of the func- tion. ( ii ) Functions do not have to be numerically values or have nu- merical inputs. Naively, it is sometimes useful to think of a function as a machine - we put the input in one end of the machine and then the machine produces an output according to the rule given by f . x f(x) f We can also represent a function geometrically. Specifically, if f is a function, we can plot all the ordered pairs ( x,f ( x )) on a coordinate plane. We call the set of points ( x,f ( x )) as x ranges over all the domain values, the graph of f ( x ). We illustrate our work so far with some examples. Example 1.2. What is the range of f ( x ) = x 2 if its domain is all x greaterorequalslant 1? 1 2 Though the algebraic range of f ( x ) = x 2 and the corresponding range is y greaterorequalslant 0, in this case we are given stronger constraints on the domain, so it is likely that stronger constraints will also exist on the range. Since x greaterorequalslant 1 and is always positive, it follows that x * x greaterorequalslant 1 * 1 = 1, so the range is y greaterorequalslant 1....
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