# Which kinds of compositions of invertible elementary and nonelementary functions are elementary?

Let $$f$$ be a bijective elementary function, elementary invertible or not.
Let $$h$$ be a bijective nonelementary function, elementary invertible or not.
Which of the compositions $$h(f(x))$$ and $$f(h(x))$$ can be elementary, and which cannot be elementary?

The elementary functions are defined in differential algebra. That are the functions $$X\in\mathbb{C}\to Y\in\mathbb{C}$$ that are composed of $$\exp$$, $$\ln$$ and/or unary or multiary univalued algebraic functions.

The elementary functions are closed regarding composition. That means, the composition of two elementary functions is elementary again.

Let $$^{-1}$$ denote the compositional inverse.

Let $$g$$ be an elementary function.

Keep in mind that $$f$$ and $$g$$ are elementary.

We formulate the problems as $$h(f(x))=g(x)$$ and $$f(h(x))=g(x)$$.

1.)

$$h(f(x))=g(x)$$

Because $$h$$ is invertible:

$$f(x)=h^{-1}(g(x))$$

If $$h^{-1}$$ is nonelementary, $$h^{-1}(g(x))$$ is elementary, and the equation is valid.

If $$h^{-1}$$ is elementary, $$h^{-1}(g(x))$$ is elementary, and the equation is valid.

A left inverse of $$h$$ is sufficient.

2.)

$$f(h(x))=g(x)$$

Because $$f$$ is invertible:

$$h(x)=f^{-1}(g(x))$$

If $$f^{-1}$$ is nonelementary, $$f^{-1}(g(x))$$ is nonelementary, and the equation is valid.

If $$f^{-1}$$ is elementary, $$f^{-1}(g(x))$$ is nonelementary only if $$g$$ is nonelementary.

A left inverse of $$f$$ is sufficient.


Functions with a left or right inverse are treated in MathStackexchange: Are elementary compositions of nonelementary functions also nonelementary?