Find two functions f (x) and g (x) such that f[g (x) ] = x but g[f (x) ] does not equal x.

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]