Using Function Notation, Worksheet 2
Homework
Write work on a separate sheet. If you think you can fit your work on this sheet, then you are not providing adequate explanation.
Use the function f(x) = 3x + 7 for all questions on this sheet.
1) Explain the difference between these two expressions: f(5) – f(3) and f(5 – 3). In each case, what steps do you go through in order to evaluate the expression?
2) Does f(5) – f(3) = f(5 – 3)?
3) See if you can find some different numbers to use in #1 to get it to work. We know that f(5) – f(3) doesn’t equal f(5 – 3) but maybe we can get f(9) – f(5) to equal f(9 – 5), or something like that. See if you can find some other pair of numbers a and b such that the value of f(a) – f(b) comes out the same as f(a – b).
If you feel that it is impossible to find anything that will work, say why you think that.
4) I know that f(34,227,885) – f(34,227,881) = 12. I figured that out very quickly, without using a calculator. How is it possible to do that?
The following questions are in the format known as "Sometimes, Always, Never." Here are some general instructions for answering that type of question.
If you believe the statement is true SOMETIMES, then you should state when it is true. Perhaps it is true only when x=0. Or perhaps it is true when x is greater than 5 or less than 2. Include all cases that make the statement true and show how you arrived at those.
If you believe the statement is true ALWAYS, then you need to present a convincing argument as to why it has to be true. There can’t be any exceptions.
If you believe the statement is true NEVER, then once again you need a convincing argument. What prevents it from being true? Are you sure there are no possible situations where it could be true?
5) When is f(-a) = -f(a)? Sometimes, Always, or Never? Remember that this refers specifically to the function that is defined by f(x) = 3x + 7. Use that for all questions.
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6) When is f(2a) = 2f(a)? Sometimes, Always, or Never?
7) When is f(a2) = (f(a))2 ? Sometimes, Always, or Never?
8) If you decided that #5, #6, and #7 are all cases of "Never", then you have made an error. Go back and check your work. See if perhaps you made a hidden assumption.
Also, you should think about how using algebra could help you with your arguments. If you haven’t used any algebra at all, you should consider that now.
Challenge Problem (Optional)
Design a function where f(a) – f(b) will come out the same as f(a – b), for all possible values of a and b. Prove that it always works.