Answer:
To find (f/g)(x), we need to divide f(x) by g(x) and simplify the result. Therefore, we have:(f/g)(x) = f(x) / g(x) = (x^2 + 2) / (x + 2)The domain of g(x) is all real numbers except -2, since division by zero is undefined. Therefore, the domain of (f/g)(x) is also all real numbers except -2.To find g(f(x)), we need to substitute f(x) into g(x) and simplify the result. Therefore, we have:g(f(x)) = (f(x))^2 + 2 = (2x^2 + 4x - 6)^2 + 2 = 4x^4 + 16x^3 - 8x^2 + 16x + 34The domain of f(x) is all real numbers, so the domain of g(f(x)) is also all real numbers.To evaluate the specified expressions, we need to substitute the given values of x into the corresponding functions and simplify the results. Therefore, we have:(fg)(2) = f(2) * g(2) = (2^2 + 2) * (2 + 2) = 24(gf)(7) = g(f(7)) = g(27^2 + 47 - 6) = g(92) = 92 + 2 = 94g(√2/7) = (√2/7)^2 + 2 = 2/49 + 2 = 2.0408...f(π/2) = 2(π/2)^2 + 4(π/2) - 6 = π^2 + 4π - 6The domain of g(x) is all real numbers, so g(207) is defined.The domain of f(x) is all real numbers, so f(4-2) and f(4/2) are defined.Therefore, the final answers are:(f/g)(x) = (x^2 + 2) / (x + 2), domain: all real numbers except -2
g(f(x)) = 4x^4 + 16x^3 - 8x^2 + 16x + 34, domain: all real numbers
(fg)(2) = 24
(gf)(7) = 94
g(√2/7) = 2.0408...
f(π/2) = π^2 + 4π - 6
g(207) is defined
f(4-2) and f(4/2) are defined.