__Appendix 3__

For those familiar with function notation in school algebra, in the situation when both sides of an equation come into play, deciding whether to use a single-letter abbreviation or not is often faced: that is, deciding whether to use *f*(*x*) notation versus the letter *y*. Once again, the viewpoint is more one of user-friendliness than of necessity.

For instance, sometimes you may see the following two representations: *f*(*x*) = 16*x* + 10 or *y* = 16*x* + 10. These both represent the same variation given by 16*x* + 10, but they indicate two different ways of looking at the situation. The first option looks at the function *f*(*x*) as being dependent in a subordinate fashion on the terms on the right-hand side. That is, we can plug in *x* = 2 to get *f*(2) = 16(2) + 10, which simplifies to 32 + 10 = 42.

However, if we want to discover what values of *x* will make 16*x* + 10 equal 106, 586, or 938, we tend to look at the second representation, *y* = 16*x* + 10. In this case, both sides of the equation are in play with equal authority. Thus, we would want to find what *x* values correspond to *y* = 106, 586, and 938, respectively. The easiest way to do this is to first solve this equation in one grand maneuver for *x*. The reduction diagram shows how this unfolds:

Now, if we place the given *y* values in the simplified equation, we get the corresponding *x* values of 6, 36, and 58. Here is the calculation of *x* when *y* = 938:

Similarly, we will obtain *x* = 6 for *y* = 106 and *x* = 36 for *y* = 586.

If we had used *f*(*x*) here instead of *y* and solved for *x*, in this notation we would have obtained

This form, though accurate, is generally less user-friendly to new learners of algebra, and probably so even to most of those well versed in algebra.

For those familiar with graphing, when we look at graphing ordered pairs of *x* and *y* on the Cartesian coordinate system, we often view the representation *y* = 16*x* + 10 as being more convenient because we have the two axes (*x*-axis and *y*-axis) in play.

Again, these are subtle viewpoints; though they are often employed in common practice, they are not absolutely necessary.