If h⁻¹(x) is the inverse of h(x), then by definition of inverse function,
h(h⁻¹(x)) = x
Given that h(x) = 6∛(2x + 5) - 1, we have
h(h⁻¹(x)) = 6∛(2 h⁻¹(x) + 5) - 1 = x
Solve for h⁻¹(x) :
6∛(2 h⁻¹(x) + 5) - 1 = x
6∛(2 h⁻¹(x) + 5) = x + 1
∛(2 h⁻¹(x) + 5) = (x + 1)/6
[∛(2 h⁻¹(x) + 5)]³ = [(x + 1)/6]³
2 h⁻¹(x) + 5 = (x + 1)³/216
2 h⁻¹(x) = (x + 1)³/216 - 5
h⁻¹(x) = (x + 1)³/432 - 5/2