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 function defined for all positive integers n, and f satisfies the two conditions above, which of the following could be the definition of f?

(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 f4 = 4-2 = 2 does not equal 4, so the second condition fails. If f(n) = 2n, then f(4) = 8 not equal to 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) = f2 = 4 and 2f1 = 24 = 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) = f2 = (22)-4 = 0 and 2f(n) = 2f1 = 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.