1. Explanation of why g is a function but does not have an inverse:

Function:
A function is a relation between a set of inputs (called the domain) and a set of outputs (called the range) in which each input is associated with exactly one output. In other words, for every input value, there is only one corresponding output value.

In the given function g(x) = 6x^4 + 5, for any real value of x, there is only one corresponding value of g(x). Hence, g(x) is a function.

Inverse:
The inverse of a function is obtained by interchanging the roles of the domain and range. In other words, if a function f(x) maps an input x to an output y, then the inverse function f^(-1)(y) maps the output y back to the input x.

To find the inverse of a function, we need to check if the function is one-to-one, which means that each output value is associated with exactly one input value. If a function is not one-to-one, it does not have an inverse.

Explanation of why g(x) does not have an inverse:
To determine if g(x) has an inverse, we need to check if g(x) is one-to-one.

Not One-to-One:
For g(x) = 6x^4 + 5, let's consider two different values of x, say x₁ and x₂, such that x₁ ≠ x₂. Now, let's calculate the corresponding values of g(x₁) and g(x₂).
g(x₁) = 6x₁^4 + 5
g(x₂) = 6x₂^4 + 5

If g(x₁) = g(x₂), then the function is not one-to-one, and thus, it does not have an inverse.

Counterexample:
Let's consider x₁ = 1 and x₂ = -1.
g(1) = 6(1)^4 + 5 = 6 + 5 = 11
g(-1) = 6(-1)^4 + 5 = 6 + 5 = 11

Since g(1) = g(-1), we have a counterexample where different inputs produce the same output. Hence, g(x) is not one-to-one, and therefore, it does not have an inverse.

2. Finding g^2(x) and stating its domain:

The notation g^2(x) represents the composition of the function g with itself. In other words, g^2(x) = g(g(x)).

Calculating g^2(x):
g(x) = 6x^4 + 5
Substituting g(x) into itself, we have:
g^2(x) = g(g(x))
= g(6x^4 + 5)
= 6(6x^4 + 5)^4 + 5

Simplifying further, we can expand (6x^4 + 5)^4 using the binomial theorem and then multiply by 6:
g^2(x) = 6(1296x^16 + 540x^12 + 90x^8 + 10x^