What's the Function?

One of my esteemed cardiologist/physiologist colleagues would begin to answer a question the same way every time: He would say - "It depends ..", and of course he was always correct.  In mathematics, one way of stating the fact that the "answer" depends on one or more variables is to express the answer as a function.   I remember my high school mathematics instructor telling us that a function was like a machine; you supply an input ( the value of one or more variables), and it gives back on output ( the value of the function ).  The function is like a prescription which states how the input variable(s) will be modified to produce an output.
Here's an example function:
\(\Large f(x)=x^2\)
For whatever reason, in mathematics people tend to use various letters to represent or name a function.  In the above, the function has been named \( f(x)\)  where the parentheses enclosing \( x\)  are to indicate explicitly that the function depends on a single variable \( x\) .  The prescription for the function is to square whatever value of \( x\)  is supplied.  In English --  I supply the function with a value for \( x\) ; the function returns the value \( x^2\) .  I'm pretty sure anyone reading this will have been exposed to some version of the above.  I hope also that you will eventually see something you haven't seen before if you keep reading.  
The answer to many mathematical (or analytical) questions is not a number ( not an algebraic result ) but a function.  A function is a relationship between dependent and independent variables.
So here's another one you're familiar with:
\( \Large y=cos(\theta)\)
This is a rather well-known trigonometric function, the cosine. You supply the function with an angle, \( \theta\) , and it returns a number equal to a ratio that has a geometric interpretation; the length of the side of a right triangle (the side adjacent to the angle in question) divided by the length of the hypotenuse.  We will soon see that there are other interpretations as to the meaning or origin of this function.  This happens to be a single valued function; for any value of \( \theta\) that you supply, there is exactly one value returned by the function with no ambiguity whatsoever.   Other important aspects to note – for this function you can supply ANY value for \( \theta\) that you like ( this will even include complex values that can have an imaginary component to them ).  The value returned will ALWAYS lie between -1.0 and 1.0 ( if the input is real ).  I'm pointing these features out because each function will have its own peculiarities
It usually makes sense to talk about the inverse of a function.  


Depends on What?

 \( \Large \sigma = p \Large\frac{h}{2r}\)



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