Problems

1. Is the function `ftext[(]xtext[)]=x^2+7` additive? Why or why not?
2. Is the function `ftext[(]xtext[)]=x^2+7` multiplicative? Why or why not?
3. Is the function `ftext[(]xtext[)]=x^2+7` even? Why or why not?
4. Is the function `ftext[(]xtext[)]=x^2+7` odd? Why or why not?
5. Is the function `phi text[(]t text[)]=t^(3//2)` additive? Why or why not?
6. Is the function `phi text[(]t text[)]=t^(3//2)` multiplicative? Why or why not?
7. Is the function `phi text[(]t text[)]=t^(3//2)` even? Why or why not?
8. Is the function `phi text[(]t text[)]=t^(3//2)` odd? Why or why not?
9. Is the function `lambda text[(]t text[)]=|t|^(3//2)` additive? Why or why not?
10. Is the function `lambda text[(]t text[)]=|t|^(3//2)` multiplicative? Why or why not?
11. Is the function `lambda text[(]t text[)]=|t|^(3//2)` even? Why or why not?
12. Is the function `lambda text[(]t text[)]=|t|^(3//2)` odd? Why or why not?
13. Here's another important functional equation:

    `f text[(]a btext[)]=f text[(]atext[)]+f text[(]btext[)]`.

    Functions that satisfy this equation turn products into sums, a useful thing to do, because sums are usually easier to work with than products. From your previous experience with algebra, you should know at least one such function. What is it? [Hint: Look again at Activity 3.] Do you know more than one such function?

14. What about functions that turn sums into products:

    `f text[(]a+btext[)]=f text[(]atext[)] f text[(]btext[)]`?

    Do you know any of those? [Hint: The operation of turning sums into products undoes the operation of turning products into sums. You know a function that turns products into sums. What undoes the effect of that function?]

15. Define what it would mean for a function to be subtractive. Can you find any functions that satisfy your definition?
16. Define what it would mean for a function to be divisive. Can you find any functions that satisfy your definition?
17. Can you think of a function that is both additive and multiplicative? Can you think of a second one?