Question

Two functions f : R → R and g : R → R are defined by f(x) = 5x3 + 1 and g(x) = 2x − 3 for all x ∈ R.

Determine the inverse of (f -1 ◦ g) and (g ◦ f )(2) and ( f ◦ g)(2).

Answer 1

$f(x)=5x^3+1\\ g(x)=2x-3\\$

Find f^-1

$x=5y^3+1\\ f^{-1}=y=\sqrt[3]{(x-1)/5}$

$f^{-1}(g)=\sqrt[3]{(2x-3-1)/5}=\sqrt[3]{(2x-4)/5}$

$g(f(2))=2(5x^3+1)-3=2(5(2^3)+1)-3=79$

$f(g(2))=5(2x-3)^3+1=5(2(2)-3)^3+1=6$