Question

What is the inverse of f(x)=(x−5)^2 for x≥5 where function g is the inverse of function f?

Answer 1

Let f(x) = y

y = (x-5)^2 
x-5 = sqrt y
x = sqrt y + 5

f-1(x) = sqrtx + 5

g(x) = sqrt x + 5     for x >= 5

Answer 2

Say, for example, that a function  f  acts on 5, producing  f(5).  Then if g is the inverse of f, then g acting on f(5) will bring back 5.

g(f(5)) = 5.

Actually, g must do that for all values in the domain of f.  And f must do that for all values in the domain of g.

In general, if a function f acts on a value of x, producing f(x),

then if g is the inverse, then g acting on f(x) -- g(f(x)) --will return x.

Here is the definition:

Functions f(x) and g(x) are inverses of one another if:

f(g(x)) = x   and   g(f(x)) = x,

for all values of x in their respective domains.