Consider the exponential function f ( x ) = e x , which has the domain I = R . From the definition, the Legendre transform is f ∗ ∗ ( x ∗ ∗ ) = sup x ∈ ∈ R ( x ∗ ∗ x − − e x ) , x ∗ ∗ ∈ ∈ I ∗ ∗ where I ∗ ∗ remains to be determined. To evaluate the supremum, compute the derivative of x ∗ ∗ x − − e x with respect to x and set equal to zero: d d x ( x ∗ ∗ x − − e x ) = x ∗ ∗ − − e x = 0. The second derivative − − e x is negative everywhere, so the maximal value is achieved at x = ln ⁡ ⁡ ( x ∗ ∗ ) . Thus, the Legendre transform is f ∗ ∗ ( x ∗ ∗ ) = x ∗ ∗ ln ⁡ ⁡ ( x ∗ ∗ ) − − e ln ⁡ ⁡ ( x ∗ ∗ ) = x ∗ ∗ ( ln ⁡ ⁡ ( x ∗ ∗ ) − − 1 ) and has domain I ∗ ∗ = ( 0 , ∞ ∞ ) . This illustrates that the domains of a function and its Legendre transform can be different.