for all integers
If is a function defined for all positive integers , and satisfies the two conditions above, which of the following could be the definition of ?
This is a tough question. Not so much because of the calculations involved or because you need to know advanced math in order to solve it. It is difficult because most people will spend a considerable amount of time trying to understand it. In a sense, this is a reading comprehension question rather than a math question.
Lets try and see what would be a good approach to this question. We should also analyze what makes it confusing and work on techniques to reduce the mess.
The questions starts with a statement:
for all integers
How could we read this information? f() means that if you place a variable inside the function will spit out what ever is written on the other side. We also note that there are limitation on what we can place in the function (also known as the Domain of the function) because we can only place n that are integers (good ! No fractions !).
What does this function do? Lets take an integer n – how about 7? If we calculate f(2 times 7 = 14) we will get 2 times the value of f(7) – whatever that maybe.
OK – not too bad – this defines our function and we understand what it can do.
What other information are we given?
OK – so when we place into the function the value 4 we get back 4.
That means that f(2 times n)=2 times f(n) and if 2 times n = 4 then n must be 2. So f(2X2)=2f(2)=4 so f(2) should be 2.
Now we are given options to chose from what f() actually does.
Since we combined the two pieces of information and concluded f(2) must give us 2 and we know f(4)=4 we can check which functions actually spit out these values when we plug in 2 or 4.
By elimination only B satisfies both conditions.