What is the inverse function of #f(x)=3x^4#?

1 Answer
Jim H
Jul 31, 2015

With its natural domain of #(-oo, oo)#, that function is not one-to-one. It does not have an inverse.

Explanation:

If we restrict the domain to some subset of #[0,oo)#, then the function has an inverse:

#y = 3x^4#

#x^4 = y/3#,

#x = root(4)(y/3)#

So the inverse function is #g(x) = root(4)(x/3)#
(Rationalize the denominator if you wish to get #g(x) = root(4)(27x)/3#.)

On the other hand:
It we restrict the domain to a subset of #(-oo, 0]#, then the inverse is #h(x) = -root(4)(x/3)#

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