General Strategy for Questions on FunctionsDate: 06/10/2004 at 09:04:57 From: Renu Subject: ACT question on functions If the function f satisfies the equation f(x + y) = f(x) + f(y) for every pair of real numbers x and y, what is (are) the possible value(s) of f(0)? A. any real number B. any positive real number C. 0 and 1 only D. 1 only E. 0 only Thanks a google! Date: 06/10/2004 at 10:15:46 From: Doctor Peterson Subject: Re: ACT question on functions Hi, Renu. There are two things I can see to do to start analyzing this functional equation to see what it says about f(0). One is to let x or y be 0 and see what happens; the other is to let x + y be zero. First, if y = 0, we know that f(x + 0) = f(x) + f(0) for any x Since x + 0 is x, this means f(x) = f(x) + f(0) for any x What does that tell you about the value of f(0)? Setting x + y to 0 (that is, taking y = -x) will give a different fact, not relevant to the question that was asked, but still interesting. This problem illustrates a useful thing to keep in mind; when you don't already know techniques for solving a problem (and functional equations don't have a lot of standard techniques!), you can just play with what you've got and see what happens. Try plugging things in and look for possibly useful implications. In other words, even when you're taking a test, you can learn just the way an infant does, putting things into his mouth and pushing them around to see if they do anything worth observing. Mathematicians are just like little kids - or rather, little kids are mathematicians in disguise! If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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