Question

g(x) = 2(x+1)^3 +8 (condition: x>1)
obtain an expression for g'(x) and use your answer to explain why g(x) has an inverse.

Answer 1

chain rule
g'(x)=h(j(x))'j'(x) h(x)=2x^3, j(x)=x+1, h'(x)=6x^2, j'(x)=1
6*(x+1)^2
6(x^2+2x+1)
If x>1 then g'(x) is always >0, then it's strictly increasing.
As g(x) is strictly increasing to all x>1, then it's inyective.
As g(x) is never 0, then g^-1(x) is defined to all x>1