For the given functions, find \( (f \circ g)(x) \) and \( (g \circ f)(x) \) and the domain of each. \( f(x)=x^{3}-8 x^{2}+2 x+7, g(x)=x+1 \) \( (f \circ g)(x)=\square \)

To find \( (f \circ g)(x) \), you substitute \( g(x) \) into \( f(x) \). So, \[ (f \circ g)(x) = f(g(x)) = f(x+1) \] Calculating this: \[ f(x+1) = (x+1)^3 - 8(x+1)^2 + 2(x+1) + 7 \] Expanding \( (x+1)^3 = x^3 + 3x^2 + 3x + 1 \) and \( (x+1)^2 = x^2 + 2x + 1 \): \[ -8(x+1)^2 = -8(x^2 + 2x + 1) = -8x^2 - 16x - 8 \] Combining all the parts: \[ f(x+1) = (x^3 + 3x^2 + 3x + 1) - (8x^2 + 16x + 8) + (2x + 2) + 7 \] \[ = x^3 + 3x^2 - 8x^2 + 3x - 16x + 2x + 1 - 8 + 2 + 7 \] \[ = x^3 - 5x^2 - 11x + 2 \] So, \[ (f \circ g)(x) = x^3 - 5x^2 - 11x + 2 \] The domain of \( (f \circ g)(x) \) is all real numbers, \( \mathbb{R} \), since both \( f(x) \) and \( g(x) \) are polynomials. Now for \( (g \circ f)(x) \): \[ (g \circ f)(x) = g(f(x)) = g(x^3 - 8x^2 + 2x + 7) \] Calculating this: \[ g(f(x)) = (x^3 - 8x^2 + 2x + 7) + 1 = x^3 - 8x^2 + 2x + 8 \] So, \[ (g \circ f)(x) = x^3 - 8x^2 + 2x + 8 \] The domain of \( (g \circ f)(x) \) is also all real numbers, \( \mathbb{R} \), since both functions are polynomials as well. In summary: \[ (f \circ g)(x) = x^3 - 5x^2 - 11x + 2 \] \[ (g \circ f)(x) = x^3 - 8x^2 + 2x + 8 \] Domains: \( \mathbb{R} \) for both.