What is the 'formula' of a composite function?

Question

Consider $f: \mathbb{R} \to \mathbb{R}$ such that $f(x) = \frac{1}{x^2 +1}$ and $g: \mathbb{R} \to \mathbb{R}\times \mathbb{R}$ given by $g(x) = (3x, x^2)$.

I was asked to find the 'formulas' of $f \circ f$ and $g \circ f$ but I am unsure what exactly it would mean by formulas in this instance? I understand that with composite functions you solve them by doing the inner function and then the outer by: $f(g(x))$ is this what I have to do in this case?

Answer 1

I don't know how to do $f(g(x))$ because $g(x)$ outputs a $2D$ point and $f$ expects a single number. To do $g(f(x))$ it is less confusing if you change the variable in $g$ to $g(y)=(3y,y^2)$, now plug in $f(x)$ for $y$, so $g(f(x))=(3(1/(x^2+1)),(1/(x^2+1))^2)$

Answer 2

If $g : \mathbb{R} \to \mathbb{R} \times \mathbb{R}$, then $g(x)$ is a $2\times 1$ vector of reals.

If $f : \mathbb{R} \to \mathbb{R}$, then the argument of $f$ is a scalar.

Hence, $(g \circ f) (x)$ makes sense, but $(f \circ g) (x)$ does not.