2004 AMC 12B Problem 13

Problem 13 of 25IntermediateAlgebra

f(x) = ax + b

and

f^{-1}(x) = bx + a

with

a

and

b

real, what is the value of

a + b?

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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 = \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.

Concepts: function · system of equations

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.