Given functions \( f \) and \( g \), find \( (a)(f \circ g)(x) \) and its domain, and \( (b)(g \circ f)(x) \) and its domain. \( f(x)=x^{3}, g(x)=x^{2}+8 x-2 \) (a) \( (f \circ g)(x)=\left(x^{2}+8 x-2\right)^{3} \) (Do not evaluate.) The domain of \( (f \circ g)(x) \) is \( \square \). (Type your answer in interval notation.)

To find \( (f \circ g)(x) \), we have: \[ f(g(x)) = f(x^2 + 8x - 2) = (x^2 + 8x - 2)^3 \] Now, we need to determine the domain of \( (f \circ g)(x) \). 1. **Domain of \( g(x) \)**: The function \( g(x) = x^2 + 8x - 2 \) is a polynomial, and polynomials are defined for all real numbers. Therefore, the domain of \( g(x) \) is \( (-\infty, \infty) \). 2. **Domain of \( f(x) \)**: The function \( f(x) = x^3 \) is also a polynomial, which is defined for all real numbers. Thus, the domain of \( f(x) \) is also \( (-\infty, \infty) \). Since \( g(x) \) is defined for all \( x \) and \( f(x) \) is defined for all real numbers, the composition \( (f \circ g)(x) \) is also defined for all real numbers. Thus, the domain of \( (f \circ g)(x) \) is: \[ \text{Domain of } (f \circ g)(x) = (-\infty, \infty) \] Now, for part (b), we need to find \( (g \circ f)(x) \) and its domain. \[ g(f(x)) = g(x^3) = (x^3)^2 + 8(x^3) - 2 = x^6 + 8x^3 - 2 \] Next, we determine the domain of \( (g \circ f)(x) \). 1. **Domain of \( f(x) \)**: As established, the domain of \( f(x) = x^3 \) is \( (-\infty, \infty) \). 2. **Domain of \( g(x) \)**: The function \( g(x) = x^2 + 8x - 2 \) is defined for all real numbers, as it is a polynomial. Since \( f(x) \) is defined for all \( x \) and \( g(x) \) is defined for all real numbers, the composition \( (g \circ f)(x) \) is also defined for all real numbers. Thus, the domain of \( (g \circ f)(x) \) is: \[ \text{Domain of } (g \circ f)(x) = (-\infty, \infty) \] In summary: (a) \( (f \circ g)(x) = (x^2 + 8x - 2)^3 \) with domain \( (-\infty, \infty) \). (b) \( (g \circ f)(x) = x^6 + 8x^3 - 2 \) with domain \( (-\infty, \infty) \).