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

Simplify the expression by following steps: - step0: Simplify: \(\left(x^{3}-7x^{2}+3x+7\right)+1\) - step1: Remove the parentheses: \(x^{3}-7x^{2}+3x+7+1\) - step2: Add the numbers: \(x^{3}-7x^{2}+3x+8\) Expand the expression \( (x+1)^{3} - 7(x+1)^{2} + 3(x+1) + 7 \) Simplify the expression by following steps: - step0: Calculate: \(\left(x+1\right)^{3}-7\left(x+1\right)^{2}+3\left(x+1\right)+7\) - step1: Expand the expression: \(x^{3}+3x^{2}+3x+1-7\left(x+1\right)^{2}+3\left(x+1\right)+7\) - step2: Expand the expression: \(x^{3}+3x^{2}+3x+1-7x^{2}-14x-7+3\left(x+1\right)+7\) - step3: Expand the expression: \(x^{3}+3x^{2}+3x+1-7x^{2}-14x-7+3x+3+7\) - step4: Subtract the terms: \(x^{3}-4x^{2}-8x+4\) Let's find \( (f \circ g)(x) \) and \( (g \circ f)(x) \) step by step. ### Step 1: Calculate \( (f \circ g)(x) \) We have: - \( f(x) = x^{3} - 7x^{2} + 3x + 7 \) - \( g(x) = x + 1 \) To find \( (f \circ g)(x) \), we substitute \( g(x) \) into \( f(x) \): \[ (f \circ g)(x) = f(g(x)) = f(x + 1) \] Calculating \( f(x + 1) \): \[ f(x + 1) = (x + 1)^{3} - 7(x + 1)^{2} + 3(x + 1) + 7 \] After expanding, we find: \[ (f \circ g)(x) = x^{3} - 4x^{2} - 8x + 4 \] ### Step 2: Calculate \( (g \circ f)(x) \) Now, we find \( (g \circ f)(x) \): \[ (g \circ f)(x) = g(f(x)) = g(x^{3} - 7x^{2} + 3x + 7) \] Calculating \( g(f(x)) \): \[ g(f(x)) = (x^{3} - 7x^{2} + 3x + 7) + 1 \] After simplifying, we find: \[ (g \circ f)(x) = x^{3} - 7x^{2} + 3x + 8 \] ### Step 3: Determine the Domains - The domain of \( g(x) = x + 1 \) is all real numbers, \( \mathbb{R} \). - The function \( f(x) \) is a polynomial, which is also defined for all real numbers, \( \mathbb{R} \). Thus, both compositions \( (f \circ g)(x) \) and \( (g \circ f)(x) \) are defined for all real numbers. ### Final Results - \( (f \circ g)(x) = x^{3} - 4x^{2} - 8x + 4 \) - \( (g \circ f)(x) = x^{3} - 7x^{2} + 3x + 8 \) - Domain of \( (f \circ g)(x) \): \( \mathbb{R} \) - Domain of \( (g \circ f)(x) \): \( \mathbb{R} \)