Maths inverse function?

g(x) = (1/2)(eˣ + e⁻ˣ) , x > 0
Let y = (1/2)(eˣ + e⁻ˣ)
2yeˣ = e²ˣ + 1
(eˣ)² - 2yeˣ + 1 = 0
Note that the sum of roots = 2y = eˣ + e⁻ˣ ≥ 2√(eˣ e⁻ˣ) = 2 ⇒ y ≥ 1 ,
y = 1 if and only if eˣ = e⁻ˣ ⇒ x = 0 is rejected, so y > 1.
eˣ = (2y ± √(4y² - 4)) / 2
eˣ = y + √(y² - 1) for x ≥ 0 or eˣ = y - √(y² - 1) for x ≤ 0 (rejected)
x = In (y + √(y² - 1)) , y > 1
∴ The inverse function of g(x) is In (x + √(x² - 1)) , x > 1 .