For the given functions \( f \) and \( g \), complete parts \( (\mathrm{a})-(\mathrm{h}) \) For parts \( (\mathrm{a}) \)-(d), also find the domain. \( f(\mathrm{x})=\mathrm{x}-1 ; g(\mathrm{x})=4 x^{2} \)

Let's analyze the functions \( f(x) = x - 1 \) and \( g(x) = 4x^2 \) and complete the required parts step by step. ### Part (a): Find \( f(g(x)) \) To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(4x^2) = 4x^2 - 1 \] ### Domain of \( f(g(x)) \) The function \( g(x) = 4x^2 \) is defined for all real numbers \( x \). Since \( f(x) \) is also defined for all real numbers, the domain of \( f(g(x)) \) is: \[ \text{Domain of } f(g(x)) = (-\infty, \infty) \] ### Part (b): Find \( g(f(x)) \) To find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(x - 1) = 4(x - 1)^2 \] ### Domain of \( g(f(x)) \) The function \( f(x) = x - 1 \) is defined for all real numbers \( x \). Since \( g(x) \) is also defined for all real numbers, the domain of \( g(f(x)) \) is: \[ \text{Domain of } g(f(x)) = (-\infty, \infty) \] ### Part (c): Find \( f(x) + g(x) \) To find \( f(x) + g(x) \): \[ f(x) + g(x) = (x - 1) + (4x^2) = 4x^2 + x - 1 \] ### Domain of \( f(x) + g(x) \) Both \( f(x) \) and \( g(x) \) are defined for all real numbers, so the domain of \( f(x) + g(x) \) is: \[ \text{Domain of } f(x) + g(x) = (-\infty, \infty) \] ### Part (d): Find \( f(x) \cdot g(x) \) To find \( f(x) \cdot g(x) \): \[ f(x) \cdot g(x) = (x - 1)(4x^2) = 4x^3 - 4x^2 \] ### Domain of \( f(x) \cdot g(x) \) Both functions are defined for all real numbers, so the domain of \( f(x) \cdot g(x) \) is: \[ \text{Domain of } f(x) \cdot g(x) = (-\infty, \infty) \] ### Summary of Parts (a)-(d) - **(a)** \( f(g(x)) = 4x^2 - 1 \), Domain: \( (-\infty, \infty) \) - **(b)** \( g(f(x)) = 4(x - 1)^2 \), Domain: \( (-\infty, \infty) \) - **(c)** \( f(x) + g(x) = 4x^2 + x - 1 \), Domain: \( (-\infty, \infty) \) - **(d)** \( f(x) \cdot g(x) = 4x^3 - 4x^2 \), Domain: \( (-\infty, \infty) \) Now, please specify parts \( (\mathrm{e}) - (\mathrm{h}) \) for further analysis.