f(x)=2x^3+3x-1 is one-to-one function on R thus f is invertible and has inverse function f-1(x)

let f^-1(4)= a, we need to find 'a'

by the definition for inverse function if f^-1(4)= a then f(a) = 4

2a^3 + 3a - 1 = 4

2a^3 + 3a = 5

we notice that for a = 1 the above equation is satisfied that is 2*1^3 + 3*1 = 5 that is f(1) = 4

f(1) = 4 follows f-1(4) = 1.