Calculate the composite functions f _º g and g _º f, and determine their domains.

To calculate (f _º g)(x) = f(g(x)), substitue into the expression for f(x) and simplify the expression to a single fraction.

Similarly, calculate (g _º f)(x) = g(f(x)) by substituting into the expression for g(x) and simplify the expression.

To obtain the domain of either f _º g or g _º f, we need to find the domain of f(x) and g(x) individually.

Since f(x) and g(x) are both rational functions, the only x-values that need to be excluded are those that make the denominator zero.

For f(x), the denominator is x⁸ + 1 and is never zero.

Since (f _º g)(x) = f(g(x)), f(x) takes as inputs the values from g(x) to obtain the domain for
(f _º g)(x), we need to intersect the domain of g(x) with the domain of the expression for
(f _º g)(x).

Now (f _º g)(x) = is a rational function so that the only x-values that need to be excluded are those that make the denominator zero or undefined. The denominator of (f _º g)(x) is x^-32+1, which is never zero.

Since (g _º f)(x) = g(f(x)), g(x) takes, as inputs, the values from f(x), to find the domain for
(g _º f)(x) we need to intersect the domain of f(x) with the domain of the expression for
(g _º f)(x).

Now (g _º f)(x) = (x⁸ + 1)⁴ is a polynomial function, which is defined for all real numbers.