1. Given f(x) = 3x^6, find f ^-1(x). Then, state whether f ^-1(x) is a function.

Question

1.   Given f(x) = 3x^6, find f ^-1(x). Then, state whether f ^-1(x) is a function.

dape · Answer

Start with substituting $f(x)$ for x and x for y, then solve for y, the y that you get is your inverse $f^{-1}(x)$.

dape · Answer

So the equation you get after the substitutions is
$$ x=3y^6 $$
Solve this for y.

anonymous · Answer

is it a function?

dape · Answer

Have you solved it?

anonymous · Answer

y = +- (x/3) ^6.

dape · Answer

That's almost right, you just get the exponent 1/6, so
$$ y=f^{-1}(x)=\pm\left(\frac{x}{3}\right)^{1/6} $$
This is not really a function since for a single value of x, you always get 2 values for the inverse (one for plus and one for minus). One can make it a function though, by ignoring either the positive or the negative value.

dape · Answer

So you could say it's a function if you look at positive x's.

anonymous · Answer

Thanks a lot