1998 AMC 12 Problem 17

Attempt Problem 17 of the 1998 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1998 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

17.

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)?

0

2

1996

1998

2000

Set

y=0

in the functional equation

The two properties determine

f(x)

directly for every real

x

Taking

y=0

gives

f(x)=x+f(0)=x+2.

Hence

f(1998)=2000,

so E is correct.

Problem 17 in other years