What is the inverse of the function as defined by f(x)=square root(x^2+4), if x>0?

Let f(x) = y

y =
sqrt(x^2+4)

We'll raise to square both sides to eliminate
the radical:

y^2 = x^2 +
4

We'll use symmetrical
property:

x^2 + 4 = y^2

We'll
keep x^2 to the left side, shifting 1 to the right
side:

x^2 = y^2 - 4

We'll take
square both sides:

x = sqrt(y^2 -
4)

Since x > 0, we'll consider only the positive
value x = sqrt(y^2 - 4).

The inverse of the given function
is f^-1(x) = sqrt(x^2 - 4).