Interpretation of composition operator when applying a function to the output of another function

Question

Refreshing my calculus skills a bit, I reviewed the chain rule:

I wondered if the composition operation $\circ$ in $g \circ f(x)$ could actually also be written as $g(f(x))$ as this would resemble how one might think about such an operation from a (functional) programming perspective.

Disclaimer

My background is in Software Engineering combined with Applied Statistical Analysis within the context of a degree in Business Administration - so unfortunately never had heavy formal training on theoretical Math. Thus apologies if I might sometimes not use the correct technical terms and/or express things a bit "unmathy".

Answer 1

I wondered if the composition operation $\circ$ in $g \circ f(x)$ could actually also be written as $g(f(x))$ as this would resemble how one might think about such an operation from a (functional) programming perspective.

Yup; in fact, that's often how function composition is introduced (having a function take another function as its input). As to which convention - $(g \circ f)(x)$ or $g(f(x))$ - to use in your own work, both are fairly accepted notations. Just be sure to remember that, in the former, the composition is usually right-to-left, or inward-out in the latter notation.

As a simple example to verify this for clarity, let $g(x) = \cos(x)$ and $f(x) = x^2$. Then $(g\circ f)(x) = g(f(x)) = \cos(x^2)$.