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The function $f(x) = k(2 - x - x^3)$ has an inverse, and $f^{-1}(3) = -2$. Find $k$.

I tried setting $f(x)$ equal to $3$ and plugging $-2$ into $x$ and I ended up with $3 = k(12)$. I'm not sure where to go from here or if this is even the correct approach.

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2 Answers 2

up vote 1 down vote accepted

$k$ is a constant, not a function; maybe the notation in the question didn't make that very clear. You literally just have to solve the equation $3=12k$.

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Ah. Yes. $k$ is a constant. That makes much more sense. Thanks! I'll accept when the time limit expires. –  tausch86 Mar 30 '13 at 23:07

That's the correct procedure to follow; now just solve for $k$, recalling $k$ is a constant.

Given $$12k = 3 \implies k = 1/4,$$ substitute into the original expression $$f(x) = k(2 - x - x^3) = \frac 14(2 - x - x^3)$$

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Actually, you can stop at $k = 1/4$! –  amWhy Mar 30 '13 at 23:12

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