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Mathematics > Standard Multiple Choice

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

for all integers

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

Right! (Applause.)

Explanation

If function f of n = n minus 2 , then function f of 4 = 4 minus 2 = 2 does not equal 4 , so the second condition fails. If function f of n = 2 times n , then function f of 4 = 8 not equal to 4 , so the second condition fails for this function also. The other three options satisfy , so it remains to check whether they satisfy the first condition.

If n = 1 , and function f of n = 4 , then function f of (2 times n) = function f of 2 = 4 and 2 times function f of 1 = 2 times 4 = 8 , so it is not true that function f of (2 times n) = 2 times function f of n for all integers . This means that the function does not satisfy the first condition. If n = 1 , and function f of n = (2 times n) minus 4 , then function f of (2 times n) = function f of 2 = (2 times 2) minus 4 = 0 and 2 times function f of n = 2 times function f of 1 = 2 times negative 2 = negative 4 , so it is not true that function f of (2 times n) = 2 times function f of n for all integers . This means that the function function f of n = (2 times n) minus 4 does not satisfy the first condition.

However, if function f of n = n , then function f of (2 times n) = 2 times n = 2 times function f of n , for all integers . Also, function f of 4 = 4 . Therefore, the function is the only option that satisfies both conditions.