Comparing and naming functions and the many uses of the equal sign A well-known math textbook series claims that there are five different uses of the = sign. The = sign occurs in Formulas, Definitions, Equations, Relations & Identities. Many students are puzzled about the differences among these seemingly comparable mathematical objects. For example · Why can I solve the equation 5x = x + 12 but not the equation 5x = 5x + 12 ? or, · Why can I solve the equation 2x = x + 2 but not the equation 2x = x + x ? or, · Why is 5x = x + 12 an equation but 5x – y = 12 a relation? or, · Why is A = pi r2 a formula but 12 = pi r2 an equation? or, · How come y = 3x + 5 is a relation but f(x) = 3x + 5 is a definition? I propose reclassifying the five objects listed above (as well as inequalities) into two classes – comparisons and assignments. Comparisons of functions When we speak of comparisons of functions we are dealing, in general, with two different functions. [In special cases, such as identities, we may be dealing with two different symbolic forms of the same function.] Equations, inequalities, relations and identities are all examples of comparisons of functions. On the other hand, when we speak of assignments we are dealing with the problem of giving a name to a single function. Formulas and definitions are examples of assigning a name to a single function. In short, assignments deal with one function – comparisons deal with two functions.[1] a. Equations Equations are comparisons of functions. When we write the equation 5x = x + 12 the equal sign does not signify that the function 5x is the same as the function x + 12. It really should be interpreted as an equal sign with a question mark overlaid [for convenience in using my word processor I will write this equal sign as ?=?]. The equation should be written 5x ?=? x + 12 and interpreted to mean “for what values of x does the function to the left of the ?=? yield the same values as the function to the right?” [2] If we look at this comparison in a graphical representation, plotting 5x and x + 12 separately on the same set of axes, the graph makes it clear that the two functions have the same value when x has the value 3. The solution set of this equation is a single point in the domain, namely {x} = {3}. Note that the solution set is the projection of the intersection of the two functions onto the domain. b.
Relations Consider the relation x + y = x2 + y2 This too is a comparison of functions – in this instance we are comparing the function of two variables x + y to the function of two variables x2 + y2 This relation should also be written using the modified equal sign, ?=? , i.e., x + y ?=? x2 + y2 and interpreted to mean “for what values of x and y does the function to the left of the ?=? yield the same values as the function to the right?” Here too plotting left and right sides separately lends a fair amount of insight into the nature of the solution set of the relation.  A bit of manipulation involving the completion of the square will show that the solution set is a circle of radius 1 divided by the square root of 2 and centered at (½,½). In analogy to the case of the solution set of the equation in one variable that we considered above, here too the solution set is the projection of the intersection of the two functions onto the domain namely {x,y} = { (x - ½)2 + (y - ½)2 = ½ }. In this case the projection of the intersection of the two functions onto the x,y plane is a circle.[3] c. Inequalities Inequalities are also comparisons of functions. The inequality 5x < x + 12 should, for pedagogic clarity, be written as 5x ?<? x + 12 and interpreted to mean “for what values of x does the function to the left of the ?< ? yield values that are smaller than the values yielded by the function to the right? Once again a plot o the left function f(x) and the right function g(x) that the solution set is immediately apparent by attending to a projection of intersecting graphs onto the domain. d. Identities Identities are also comparison of functions – although in this case the functions on either side of the comparison operator are simply different symbolic forms of the same function. For pedagogic clarity identities might be written as 2x == x + x and interpreted to mean that the functions on the left and the function on the right yield the same values for all values of the variable x. Assigning names to functions As we stated earlier, assignments deal with the problem of naming a single function. Both formulas and definitions are examples of assignments. a. Formulas When we write the formula for the area of a circle Area of circle: A = pi r2 we are giving the reader a recipe or a prescription for calculation. The name of this recipe is A. The formula, for pedagogic clarity should be written as A := pi r2 and interpreted to mean “A is the name of the calculational recipe pi r2”. b. Definitions If we think of the equation 5x ?=? x + 12 as a comparison of two functions we might want to give the functions on each side of the ?=? a name. Suppose we wanted to call the function on the left f(x) and the function on the right g(x). Then f(x) := 5x and g(x) := x + 12. These are not equations – we cannot ask the crucial question that we must ask of equations - “for what values of x does the function on the left yield the same values as the function on the right?”. Rather f(x) and g(x) are names for 5x and x + 12 respectively. Our equation can be written as f(x) ?=? g(x) Confounding comparisons and assignments How then are we to make sense of the mathematical statement y = 2x + 5 which we find commonly written in algebra textbooks? Is this to be regarded as the comparison of two functions of two variables, i.e. f(x,y) ?=? g(x,y) where f(x,y) := y and g(x,y) := 2x + 5 or is this to be regarded as an assignment – one in which we assign the name y to the function of a variable x, i.e., y := 2x + 5 The ambiguity seems less severe if we allow for the possibility of each side of the mathematical statement to contain both variables. For example consider the statement (x + y)3 = 2x + 5. If we wish to hold onto the notion of assignment here we must introduce notions of explicit and implicit definitions of functions. Note that such a statement may in general be simultaneously an implicit definition of y(x) and/or x(y). On the other hand, seen from the perspective of comparison the alteration we made produces no conceptual difficulty – we are simply comparing two functions each of which depends on two variables. This confounding of comparison and assignment also produces the ambiguous interpretation we make of the plane of the paper on which we plot graphs. Is the y axis to be interpreted as the one along which we plot the values in the range of a function of one variable or as an independent variable y which together with x constitutes the domain of the two functions f(x,y) := 2x + 5 and g(x,y) := y When plotted as functions of two variables each of these functions is a plane and their intersection is a line – whose projection onto the x,y plane is the conventional graph of y = 2x + 5. A closing remark on rules for manipulating comparisons of functions If students explore comparisons of functions in graphical environments, plotting the functions on left and right side of separately on the same set of axes, then the nature of the solution set of the comparison is immediately evident. Moreover, all the rules for manipulating functions symbolically can now be replaced by a single rule that appeals to the students’ direct visual access to the graphical representation – i.e., Any symbolic manipulation that leaves the solution set unchanged is a legitimate symbolic action. [1] This distinction is reflected in the symbolic manipulation problems we ask students to work on. Problems of the form “factor”, “expand”, “simplify” “collect terms”, etc. are all problems that require the student to rewrite a single function in an equivalent symbolic form. Problems of the form “solve” require the student to work on (one or more) comparison(s) of functions. [2] The set of values of x for which the function on the left and the function on the right have the same value is called the solution set of the equation. [3] In the case of an inequality sign the solution set is either a circular disc on the {x,y} plane or its complement depending of the direction of the inequality. Judah L. Schwartz
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