if $$f(x)=kx^3-1 $$ and $$f ^{-1}(15)=2$$, find k.

Question

whpalmer4 · Answer

What does $f^{-1}(15) = ?$

anonymous · Answer

= 2 sorry!

whpalmer4 · Answer

I would figure out the formula for $f^{-1}(x)$ in the usual fashion of swapping x and y and then solving for y.  Then plug in the known value and solve for k.

anonymous · Answer

$$f^{-1}(15)=\frac{ 1 }{ kx^3-1 }$$
$$2=\frac{ 1 }{ k(15)^3-1 }$$

whpalmer4 · Answer

@Azteck that's supposed to be the inverse function, I think, not the reciprocal of f(x).

anonymous · Answer

Oh crap. yeah sorry about that. It's meant to be this.
$$f^{-1}(15)=\sqrt[3]{\frac{ y-1 }{ k }}$$

anonymous · Answer

x-1 sorry*

anonymous · Answer

not y-1

whpalmer4 · Answer

Too bad it isn't $f(x) = 1/x$ as $f^{-1}(x) = 1/f(x)$ in that case :-)

anonymous · Answer

oh okay so uhmm... i got $$f^{-1}(x) = \sqrt[3]{\frac{ x+1 }{ 3 }}$$

anonymous · Answer

Yeah that's right @alexeis_nicole I made a mistake on not changing the sign.

anonymous · Answer

sorry... i meant k not 3 as a denominator

anonymous · Answer

$$2=\sqrt[3]{\frac{ 15+1 }{ k }}$$
$$8=\frac{ 15+1 }{ k }$$
$$8k=16$$
$$k=2$$

anonymous · Answer

@whpalmer4 @Azteck THANK YOU !

anonymous · Answer

You're welcome.

whpalmer4 · Answer

I tell you how to do it, and point out when the other guy is leading you astray, but he gets the medal.  Harrumph :-)

whpalmer4 · Answer

Even if @Azteck hadn't given me a medal, I'd still be a happy camper, as you two gave me the idea of finding the function whose inverse is its reciprocal, don't think I ever noticed that before!