Function names in C++: Capitalize or not?

There isn't a 'correct way'. They're all syntactically correct, though there are some conventions. You could follow the Google style guide, although there are others out there.

From said guide:

Regular functions have mixed case; accessors and mutators match the name of the variable: MyExcitingFunction(), MyExcitingMethod(), my_exciting_member_variable(), set_my_exciting_member_variable().

The most common ones I see in production code are (in this order):

myFunctionName     // lower camel case

MyFunctionName     // upper camel case

my_function_name   // K & R ?

I find the naming convention a programmer uses in C++ code usually has something to do with their programming background.

E.g. ex-java programmers tend to use lower camel case for functions

Since C++11, you may want to use either snake_case or camelCase for function names.

This is because to make a class work as the range-expression in a range-based for-loop, you have to define functions called begin and end (case-sensitive) for that class.

Consequently, using e.g. PascalCase for function names means you have to break the naming consistency in your project if you ever need to make a class work with the range-based for.

Most code I've seen is camelCase functions (lower case initial letter), and ProperCase/PascalCase class names, and (most usually), snake_case variables.

But, to be honest, this is all just guidance. The single most important thing is to be consistent across your code base. Pick what seems natural / works for you, and stick to it. If you're joining a project in progress, follow their standards.