The function f R 0 defined as f x ex31 is bijective Find i

The function f : R (0,) defined as f (x) = e^(x^3+1) is bijective. Find its inverse

Solution

Given that

f(x) = e^{(x^3+1)}

Let f(x) = e^{(x^3+1)} = y

      e^{(x^3+1)} = y

Suppose f : A B is a bijection. Then the inverse of f, denoted by f ¹ : B A, is the function denoted by the rule

f(x) = y then x = f^-1(y)

If a function mapping from x to y then the inverse function of f(x) maps back y to x

e^{(x^3+1)} = y

   ln( e^{(x^3+1)} ) = ln(y) [since,if f(x)=g(x) then ln(f(x)) = ln(g(x)) ]

(x^3 + 1) ln(e) = ln(y) [ since, log a^m = m log a ]

(x^3 + 1 ).1 = ln(y) [ since , ln(e) = 1 ]

   (x^3 + 1 ) = ln(y)

x^3 = ln(y) - 1

x = ( ln(y) - 1 )^1/3

Substitute y = x , x=y

y =  ( ln(x) - 1 )^1/3

Hence,

f^-1(x) =  ( ln(x) - 1 )^1/3