每日一题

Mathematics>Standard Multiple Choice

Read the following SAT test question and then click on a button to select your answer.

f(2n) = 2f(n) for all integers n
f(4) = 4
If f is a defined for all positive integers n, and f satisfies the two conditions above, whichthe following could be the definitionf?

(A) f(n)= n-2
(B) f(n)= n
(C) f(n)= 2n
(D) f(n)= 4
(E) f(n)= (2n)-4

答案和解析

答案：B

解析：

If f(n) = n - 2, then f(4) = 4 - 2 = 2 != 4，so the second condition fails. If f(n) = 2n， then f(4) = 8!=4，so the second condition fails for this function also. The other three options satisfy f(4) = 4, so it remains to check whether they satisfy the first condition.
If n = 1, and f(n) = 4, then f(2n) = f(2) = 4 and 2f(1) = 2(4) = 8，so it is not true that f(2n)= 2f(n) for all integers n. This means that the function f(n) = 4 does not satisfy the first condition. If n = 1, and f(n) = 2n - 4, then f(2n) = f(2) = 2(2) - 4 = 0 and 2f(n) = 2f(1) = 2( -2) = -4, so it is not true that f(2n) =2f(n) for all integers n. This means that the function f(n) = 2n - 4 does not satisfy the first condition.
However, if f(n) = n, then f (2n) = 2n = 2f (n)，for all integers n. Also, f (4) = 4 • Therefore, the function f(n) = n is the only option that satisfies both conditions.