2004 AMC 12B Problem 13

Attempt Problem 13 of the 2004 AMC 12B 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 2004 AMC 12B solutions, or check the answer key.

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13.

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

Use

f(f^{-1}(x)) = x

Expanding

a(bx + a) + b = x

gives

ab = 1

and

a^2 + b = 0

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 = \frac{1}{a}

and

a^2 + \frac{1}{a} = 0,

a^3 = -1,

giving

a = -1

and

b = -1.

Thus

a + b = -2.

Thus, the correct answer is A.

Problem 13 in other years