Here is one that often comes up in inverse function discussions. People tend to take these as inverses, but they are not.
f(x) = x², domain (-∞, ∞)
g(x) = √(x), domain [0, ∞)
f[g(x)] = x, for x on the domain of g
g[f(x)] = |x|, for x on the domain of f
g[f(x)] = x only for x 0n [0, ∞)
Here is another pair that are often incorrectly taken as inverses.
f(x) = sin(x), domain (-∞, ∞)
g(x) = sin⁻¹(x), domain [-1, 1]
f[g(x)] = x, for x on the domain of f
g[f(x)] = x only for x on [-π/2, π/2]