2004 AMC 12B Problem 13

Below is the professionally curated solution for Problem 13 of the 2004 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 12B solutions, or check the answer key.

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Concepts:function system of equations

Difficulty rating: 1580

13.

If $f(x) = ax + b$ and $f^{-1}(x) = bx + a$ with $a$ and $b$ real, what is the value of $a + b?$

$-2$

$-1$

$0$

$1$

$2$

Solution:

Since $f(f^{-1}(x)) = x,$ we have $a(bx + a) + b = x.$ Matching terms gives $ab = 1$ and $a^2 + b = 0.$ Then $b = 1/a$ and $a^2 + 1/a = 0,$ so $a^3 = -1,$ giving $a = -1$ and $b = -1.$ Thus $a + b = -2.$

Thus, the correct answer is A.

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