Difference between revisions of "2004 AMC 12B Problems/Problem 13"

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Revision as of 18:57, 3 July 2013

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 $a+b = -2 \Rightarrow \mathrm{(A)}$.

See also

2004 AMC 12B (ProblemsAnswer KeyResources)
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
All AMC 12 Problems and Solutions

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