Boolean formula vs Boolean function

I’m having trouble seeing the difference between a boolean formula and a boolean function, can you please highlight the difference?

A bit of philosophy: What is the difference between a dog, a “photo of a dog”, and the string of three letters “dog”?
These are the object, a depiction of the object, and a representation of the object using the English alphabet.

A function is an abstract definition.
A formula is a representation of a function.

Think of linear algebra. You have linear transformations and matrices.
A matrix is a representation of a linear transformation.

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