What denotes the essence of a function object in mathematics? In other words, when does something become a function, and why? Take this, for example:

x = y (x + z) = 350


Is anything enclosed within the brackets considered an input to the function?

What if expressed this way:

x = y + xz = 350


Where is the function? What purpose does a function serve to measure an input and output when an input and output do not need a function? You would just employ the equivalent operator, a few variables/constants, and express a result.

What I'm asking is, why employ a function, and what is a function overall?
What you are describing is an equality.  For it to be a function you would have to specify what inputs are possible and what outputs you are looking for.  In your equality above you might say: if x is 2 and y is 3, what is z?  Now you are headed towards having a function.  If you say, z is defined by this equality where x and y are real numbers, you do have a function.  It takes certain inputs (real numbers) and spits out an output which is the real number z (whether you are actually able to compute z or not, which is a different issue).

By thinking of functions as input - operation -output we gain an immense amount -- almost impossible to discuss briefly.  We use mathematical notation like f(x,y) = z to express that x and y are inputs, f defines the operation and z is the output.  There are plenty of other ways to notate functions, and many of them are in constant use.  But this particular way of notating them has proved very fruitful.

There are several purposes to notation.  One is to express thoughts in a succint way, so that you don't need a 100 word essay to describe what you are talking about.  Compare f(x,y) = z with  "x and y are inputs, f defines the operation and z is the output".

Another feature of "fruitful" is that the notation should make it easier to understand what is being supposed, or discussed or is to be proved.

Yet another is that it organizes the information.  It's easy enough to make mistakes, and it can be very difficult to track them down if you have 18 pages of general disorganized junk (actually it is hard enough when things are organized).

Another is that it proves to be a useful way to write things in a very large number of disparate problems.

Most fruitful of all is when the notation not only accomplishes the above but is suggestive of what you might look for, or how you might prove things. It is so good it puts you onto truths that you might never suspect had not the notation pointed that way.

The notation y = f(x) (and its generalizations) certainly meet all the criteria above.
What you are describing is an equality.  For it to be a function you would have to specify what inputs are possible and what outputs you are looking for.  In your equality above you might say: if x is 2 and y is 3, what is z?  Now you are headed towards having a function.  If you say, z is defined by this equality where x and y are real numbers, you do have a function.  It takes certain inputs (real numbers) and spits out an output which is the real number z (whether you are actually able to compute z or not, which is a different issue).

By thinking of functions as input - operation -output we gain an immense amount -- almost impossible to discuss briefly.  We use mathematical notation like f(x,y) = z to express that x and y are inputs, f defines the operation and z is the output.  There are plenty of other ways to notate functions, and many of them are in constant use.  But this particular way of notating them has proved very fruitful.

There are several purposes to notation.  One is to express thoughts in a succint way, so that you don't need a 100 word essay to describe what you are talking about.  Compare f(x,y) = z with  "x and y are inputs, f defines the operation and z is the output".

Another feature of "fruitful" is that the notation should make it easier to understand what is being supposed, or discussed or is to be proved.

Yet another is that it organizes the information.  It's easy enough to make mistakes, and it can be very difficult to track them down if you have 18 pages of general disorganized junk (actually it is hard enough when things are organized).

Another is that it proves to be a useful way to write things in a very large number of disparate problems.

Most fruitful of all is when the notation not only accomplishes the above but is suggestive of what you might look for, or how you might prove things. It is so good it puts you onto truths that you might never suspect had not the notation pointed that way.

The notation y = f(x) (and its generalizations) certainly meet all the criteria above.