Part 1 of 4 Forthe given functions, find \( (f \circ g)(x) \) and \( (g \circ f)(x) \) and the domain of each. \( f(x)=x^{3}-7 x^{2}+3 x+7, g(x)=x+1 \) \( (f \circ g)(x)=\square \)

To find \( (f \circ g)(x) \), we begin by substituting \( g(x) = x + 1 \) into \( f(x) \): \[ (f \circ g)(x) = f(g(x)) = f(x + 1) \] Now, substitute \( x + 1 \) into \( f(x) \): \[ f(x + 1) = (x + 1)^3 - 7(x + 1)^2 + 3(x + 1) + 7 \] Calculating each term: 1. \( (x + 1)^3 = x^3 + 3x^2 + 3x + 1 \) 2. \( -7(x + 1)^2 = -7(x^2 + 2x + 1) = -7x^2 - 14x - 7 \) 3. \( 3(x + 1) = 3x + 3 \) 4. Finally, there’s a constant \( + 7 \). Combining these gives: \[ f(x + 1) = (x^3 + 3x^2 + 3x + 1) + (-7x^2 - 14x - 7) + 3x + 3 + 7 \] Now simplifying: \[ = x^3 + (3 - 7)x^2 + (3 - 14 + 3)x + (1 - 7 + 3 + 7) \] \[ = x^3 - 4x^2 - 8x + 4 \] Thus, we have: \[ (f \circ g)(x) = x^3 - 4x^2 - 8x + 4 \] Now, for \( (g \circ f)(x) \): \[ (g \circ f)(x) = g(f(x)) = g(x^3 - 7x^2 + 3x + 7) = (x^3 - 7x^2 + 3x + 7) + 1 \] Simplifying gives: \[ (g \circ f)(x) = x^3 - 7x^2 + 3x + 8 \] Next, let’s identify the domains: The function \( g(x) = x + 1 \) is defined for all real numbers \( x \), so its domain is \( \mathbb{R} \). The function \( f(x) = x^3 - 7x^2 + 3x + 7 \) is also defined for all real numbers, making its domain \( \mathbb{R} \) as well. Therefore, both compositions \( (f \circ g)(x) \) and \( (g \circ f)(x) \) inherit this domain: \[ \text{Domain of } (f \circ g)(x) = \mathbb{R} \quad \text{Domain of } (g \circ f)(x) = \mathbb{R} \]