Some Properties a Function Can Have
NOTE: You will need blank sheets as scratchpaper to experiment/think!

1. Recall that for a function g(x), g(2·x) is not the same thing as 2·g(x) For example, if g(x)=3x+5 then we can deduce direct formulas for functions which are defined in terms of this g(x), for example:

   1. If h(x) = g(2·x) then h(x) = ____________________

   2. While if f(x) = 2·g(x) then f(x) = ____________________

2. Let's say that a function g(x) has property (P2) if g(x) is special enough that the two expressions above are equal, that is, if g(2·x) is, for this particular function g(x), the same as 2·g(x) [Remember, saying two expressions-in-x are "equal" is to say they equal one another, as numbers, for any value we plug in for x]
   1. Show that if f(x) = x² then f(x) does not have property (P2).
      Warning: We're applying the (P2) definition to a function called "f" (not "g") now!

      Find at least one other function that does not have property (P2): __________

      Find at least one function that does have property (P2).

3. Define "Property (P3)" in a reasonable (analogous!) way and then go through parts (i) through (iii) of the preceding question, for this property.

4. How do the functions f(x)=x, f(x)=x+1, f(x)=0, and f(x)=1 behave in terms of (P2)? In terms of (P3)? How do you think they would behave with respect to property (P4)? (P5)? Can you define a "Property (Pk)" where you leave "k" as is -- as an unknown constant integer? How much of this discussion do you think would work for this general "property (Pk)"?

5. For concreteness, let's go back to (P2). There is a connection to proportionality. Can you get at this connection in a well phrased, accurate, and precise mathematical summary? Before you try to write a paragraph, go through these steps:
   1. Let s(x) be the function which outputs the area of the square whose side has length x. Let r(x) be the function which outputs the area of the rectangle of length x and of width 7. First, without writing, using, or thinking anything about formulas for these functions, argue why one of them has property (P2) and the other does not. You will need to turn the formalities of property (P2) into everyday language. Then use a non-functional way to look at s(x) and r(x); for example, draw pictures and convince yourself, then a partner, why either of the functions here does or does not has (P2), using geometric reasoning.

   2. Now, do write down formulas for s(x) and r(x), and confirm your geometric reasoning by verifying algebraically that for each of these two functions, that it does or does not have property (P2), as you claimed. Remember that to have the property, the (P2)-equation in question must work for that function's formula, for all x. To not have it, means that for at least one value of x, the corresponding equation fails.

   3. Repeat the above two questions (i) and (ii), with (P2) replaced by (P5).

6. Draw geometrically what (P2) looks like. Start by drawing the x-axis and y-axis, and label "x" and "2x" where they belong. Then label "f(x)" and "2·f(x)" where they should go. What does the picture tell you? What if you drew 4x and 4·f(x) on the appropriate axis, respectively, where they belong?

7. What if a function f(x) has the property (Pk) for all integers k? Then, if f(x) is continuous, it will obey (Pk) for all real numbers k. What can you conclude about such an f(x)? Well f(x) = f(x·1) = x· [..you fill in the rest...] What kind of function must f(x) be?