2004 AMC 12B Problems/Problem 13

Problem

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

$\mathrm{(A)}\ -2 \qquad\mathrm{(B)}\ -1 \qquad\mathrm{(C)}\ 0 \qquad\mathrm{(D)}\ 1 \qquad\mathrm{(E)}\ 2$

Solution

By the definition of an inverse function, $x = f(f^{-1}(x)) = a(bx+a)+b = abx + a^2 + b$ . By comparing coefficients, we have $ab = 1 \Longrightarrow b = \frac 1a$ and $a^2 + b = a^2 + \frac{1}{a} = 0$ . Simplifying, $a^3 + 1 = 0$ and $a = b = -1$ . Thus $a+b = -2 \Rightarrow \mathrm{(A)}$ .

2004 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2004 AMC 12B Problems/Problem 13

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