Explanation:
**(a) f(g(x))**
**g(x) = √(x²+2x+6)**
**f(g(x)) = f(√(x²+2x+6))**
**f(x) = 8x-3**
**f(√(x²+2x+6)) = 8√(x²+2x+6)-3**
Therefore, f(g(x)) = 8√(x²+2x+6)-3
**(b) g(f(x))**
**f(x) = 8x-3**
**g(f(x)) = g(8x-3)**
**g(x) = √(x²+2x+6)**
**g(8x-3) = √((8x-3)²+2(8x-3)+6)**
**g(8x-3) = √(64x²-48x+9+16x-6+6)**
**g(8x-3) = √(64x²-32x+9)**
Therefore, g(f(x)) = √(64x²-32x+9)
**(c) f(f(x))**
**f(x) = 8x-3**
**f(f(x)) = f(8x-3)**
**f(8x-3) = 8(8x-3)-3**
**f(8x-3) = 64x-24-3**
**f(8x-3) = 64x-27**
Therefore, f(f(x)) = 64x-27
**(d) g(g(x))**
**g(x) = √(x²+2x+6)**
**g(g(x)) = g(√(x²+2x+6))**
**g(x) = √(x²+2x+6)**
**g(√(x²+2x+6)) = √((√(x²+2x+6))²+2(√(x²+2x+6))+6)**
**g(√(x²+2x+6)) = √((x²+2x+6)+2√(x²+2x+6)+6)**
**g(√(x²+2x+6)) = √(x²+2x+6+12+6)**
**g(√(x²+2x+6)) = √(x²+2x+24)**
Therefore, g(g(x)) = √(x²+2x+24)
**(e) f(g(-2))**
**g(-2) = √((-2)²+2(-2)+6)**
**g(-2) = √(4-4+6)**
**g(-2) = √(6)**
**f(g(-2)) = f(√(6))**
**f(x) = 8x-3**
**f(√(6)) = 8(√(6))-3**
**f(√(6)) = 8√(6)-3**
Therefore, f(g(-2)) = 8√(6)-3
**(f) f(f(-5))**
**f(-5) = 8(-5)-3**
**f(-5) = -40-3**
**f(-5) = -43**
**f(f(-5)) = f(-43)**
**f(x) = 8x-3**
**f(-43) = 8(-43)-3**
**f(-43) = -344-3**
**f(-43) = -347**
Therefore, f(f(-5)) = -347
bardAI