How do you determine whether the function #f(x)=x^4# has an inverse and if it does, how do you find the inverse function?

In order to have an inverse function, a function must be one to one.

In the case of #f(x) = x^4# we find that #f(1) = f(-1) = 1#. So #f(x)# is not one to one on its implicit domain #RR#.

If we restrict the domain of #f(x)# to #[0, oo)# then it does have an inverse function, namely:

#f^(-1)(y) = root(4)(y)#

Some more details...

In general, if you want to find the inverse of a function #f(x)# then set #y = f(x)# and attempt to solve for #x#.

In our example, put:

#y = f(x) = x^4#

Then taking the square root of both ends, allowing for both possible signs we find:

#+-sqrt(y) = x^2#

Assuming that we are only interested in real values of #x#, we require the left hand side to be non-negative, so assume the non-negative square root.

Then take the square root of both sides again to get:

#+-sqrt(sqrt(y)) = x#

That is:

#x = +-sqrt(sqrt(y)) = +-root(4)(y)#

Note that this does not give us a unique value of #x# in terms of #y#.

So the inverse function is not defined, unless we have a restriction such as #x >= 0#, i.e. restrict the domain of #f(x)# to be #[0, oo)#.