Originally Posted by

**emakarov** Function composition is denoted by $\displaystyle f\circ g$, or f o g in plain text. Some sources may skip the circle, but I would expect them to note this convention explicitly. In any case, it is much better to write a bit more explanations than to let people wonder about notation.

You showed that $\displaystyle (f\circ g)(x) = x$, which means that g is the (right) inverse of f. This is the relationship between the functions. There are many other relationships, e.g., $\displaystyle (g\circ f)(x) = x$, $\displaystyle f(x) = (g\circ g)(x)$, $\displaystyle g(x)=(f\circ f)(x)$, $\displaystyle f(x)=1-\frac{1}{g(x)}$, $\displaystyle g(x)=f(x+1)+1$, etc., but apparently they are not relevant to the question (except maybe the first).

Note also that both sides of equality should have the same type: either functions or real numbers. For example, $\displaystyle f\circ g$ is a function (the composition of f and g), while x is a number. So, writing $\displaystyle f\circ g=x$ is not the best. One should write $\displaystyle (f\circ g)(x)=x$ or $\displaystyle f\circ g=\mathrm{id}$ where $\displaystyle \mathrm{id}(x)=x$ is the identity function.

Since $\displaystyle f(g(x))=x$, you can cancel 1994 applications $\displaystyle f(g(\dots))$ in $\displaystyle f^{2011}g^{1994}(1/2)$ and get $\displaystyle f^{2011-1994}(1/2)$. In evaluating this, note also that $\displaystyle f^3(x)=x$ (this can be verified directly; it also follows from $\displaystyle g=f^2$ and $\displaystyle f\circ g=\mathrm{id}$).