Determine the inverse function of f(x)=2x/2(x^3+x) .

For the beginning, we'll simplify the fraction that
represents the expression of the function. We'll factorize by x the
denominator:

f(x) = 2x/2x(x^2 +
1)

We'll simplify and we'll
get:

f(x) = 1/(x^2 + 1)

We'll
write f(x) = y.

y = 1/(x^2 +
1)

We'll change y by x:

x =
1/(y^2 + 1)

We'll multiply by 1/(y^2 + 1) both
sideS:

x(y^2 + 1) = 1

We'll
remove the brackets:

xy^2 + x =
1

We'll subtract x both
sides:

xy^2 = 1 - x

We'll
divide by x:

y^2 =
(1-x)/x

We'll take radicals both
sides:

sqrt y^2 =
sqrt[(1-x)/x]

y =
sqrt[(1-x)/x]

The inverse function is f^-1(x)
= sqrt[(1-x)/x].