Solution: Our first task is to identify the dependent and independent variables. Practice Problem: Define a function that expresses the area of a square in terms of the length of one of its sides. That is, we only deal with real variables and real-valued functions. ![]() We will deal only with functions whose dependent and independent variables take values in the set of real numbers only. An example of an algebraic relation that is not a function is g( x), given below.įor a given value of x, g can be either a positive or negative value hence, g is not a function. Note that in our example above, f( x) is a function because regardless of the value we choose for x, f can only be one value (12 times x). If, however, for every possible value of the independent variable, the relation has only one corresponding value for the dependent variable, then the relation is called a function. This type of "formula" is generally called an algebraic relation. The formula tells us that 10 feet is equivalent to 120 inches. Now, to convert any number of feet into inches, we just plug the number of feet into our formula in place of the independent variable. What we've done here is give some mathematical rigor to the intuitive process of converting feet into inches. Since the dependent variable depends (as the name indicates) on the independent variable, we can list x in parentheses next to f, indicating that f depends on x: Concomitantly, we call x the independent variable, since our choice of x is (essentially) arbitrary. Because the value of f depends on the value of x, we call f in this case the dependent variable. Note that as the variable x varies, so does f. Furthermore, we might give this formula its own symbolic name-for instance, f. Here, x is an undefined number called a variable. More generally, we can write a "formula" for converting x feet into inches. If the measurement is 2 feet, then we multiply by 12 to get the number of inches. For instance, we might want to convert a length measurement from feet to inches. In many situations in everyday life, we convert one set of numbers (which might even just be a single number) into other sets of numbers by performing some series of mathematical operations on it. ![]() Plot coordinate graphs of simple functions.
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