Remark on rules for manipulating comparisons of functionsĬomparisons of functions in graphical environments, plotting the functions on Line – whose projection onto the x,y plane is the conventional Two variables each of these functions is a plane and their intersection is a Y which together with x constitutes the domain of the two functions In the range of a function of one variable or as an independent variable Is the y axis to be interpreted as the one along which we plot the values The plane of the paper on which we plot graphs. Seen from the perspective of comparison the alteration we made produces noĬonceptual difficulty – we are simply comparing two functions each of whichĬomparison and assignment also produces the ambiguous interpretation we make of Simultaneously an implicit definition of y( x) and/or x( y). Note that such a statement may in general be Notion of assignment here we must introduce notions of explicit and implicitĭefinitions of functions. Severe if we allow for the possibility of each side of the mathematical statement f(x,y) ?=? g(x,y) where f(x,y) := y and g(x,y) := 2 x + 5 or is this to be regarded as anĪssignment – one in which we assign the name y to the function of a variable x, Regarded as the comparison of two functions of two variables, i.e. + 5 which we find commonly written in algebra textbooks? Is this to be The same values as the function on the right?”. These are not equations – we cannot ask theĬrucial question that we must ask of equations - “ for what values of x does the function on the left yield Suppose we wanted to call the function on the left f(x) and the function on the right g(x). Might want to give the functions on each side of the ?=? a name. If we think of the equation 5 x ?=? x + 12 as a comparison of two functions we The formula, for pedagogic clarity should be written as A := pi r 2Īnd interpreted to mean “A is the name of the calculational recipe pi r 2”. The area of a circle Area of circle: A = pi r 2 weĪre giving the reader a recipe or a prescription for calculation. Both formulasĪnd definitions are examples of assignments. Right yield the same values for all values of the variable x.Īssignments deal with the problem of naming a single function. Interpreted to mean that the functions on the left and the function on the Identities might be written as 2 x = x + x and The comparison operator are simply different symbolic forms of the same Solution set is immediately apparent by attending to a projection ofĬomparison of functions – although in this case the functions on either side of Right? Once again a plot o the left function f(x) and the right function g(x) that the Yield values that are smaller than the values yielded by the function to the The inequality 5 x < x + 12 should,įor pedagogic clarity, be written as 5 x ? < ? x + 12 and interpreted to In this case the projection of the intersection of the twoįunctions onto the x, y plane is a circle. The solution set of this equation is a single point in the domain, namely. Two functions have the same value when x has Of axes, the graph makes it clear that the In a graphical representation, plotting 5 x Processor I will write this equal sign as ?=?]. It really should be interpreted as anĮqual sign with a question mark overlaid [for convenience in using my word In short, assignments deal with one function –įunctions. Formulas and definitions are examples of assigning a name When we speak of assignments we are dealing with the problem of giving a name Identities are all examples of comparisons of functions. Symbolic forms of the same function.] Equations, inequalities, relations and [In special cases, such as identities, we may be dealing with two different Of functions we are dealing, in general, with two different functions. How come y = 3 x + 5 is a relation but f(x) = 3 x + 5įive objects listed above (as well as inequalities) into two classes – comparisons.Why is 5 x = x + 12 an equation but 5 x – y = 12 a relation? or,.Solve the equation 2 x = x + 2 but not the equation 2 x = x + Solve the equation 5 x = x + 12 but not the equation 5 x = 5 x + 12 ? or , Students are puzzled about the differences among these seemingly comparable Occurs in Formulas, Definitions, Equations, Relations & Identities.
#EQUAL SIGN SERIES#
Series claims that there are five different uses of the = sign. And naming functions and the many uses of the equal sign
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