1998 AHSME Problem 17

Number Theory Level 1

Let f(x) be a function with the two properties: (a) for any two real numbers x and y, f(x+y) = x + f(y), and (b) f(0) = 2.

What is the value of f(1998)?

2000 1996 2 1998

Daniel Hirschberg by Daniel Hirschberg , 39, USA

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3 solutions

f ( 1998 ) = f ( 1998 + 0 ) = 1998 + f ( 0 ) = 1998 + 2 = 2000 f(1998) = f(1998 + 0) = 1998 + f(0) = 1998 + 2 = \boxed{2000}

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Ajay Maheshwari
May 30, 2014

According to question if f(0) then the o/p is 2 so if we pass 1998 in function that means 2 are add in output so the answer of f(1998)= 1998+2=2000

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we have f(x+y)=x+f(y) & f(0)=2; by substituting y=0 in f(x+y) we get f(x)=x+2 and by substituting x=1998 in last equation we get result as 2000

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