May 10th, 2014, 07:18 AM  #1 
Senior Member Joined: Feb 2014 From: Louisiana Posts: 156 Thanks: 6 Math Focus: algebra and the calculus  Functional notation
Although functional notation is easy to understand, there are still some ambiguities to me. If $\displaystyle y = f(x)$ reads "y is a function of x," then what exactly does the $\displaystyle f$ symbolize in a typical function like $\displaystyle f(x) = 2x + 1$? Does $\displaystyle f$ represent the rule of correspondencethat is, multiplying by 2 and adding 1 or does it represent a variable quantity, like y? Or both? When people describe functions, they usually that $\displaystyle x$ "goes through" $\displaystyle f$ to produce a value of $\displaystyle f(x)$, but this only adds to the ambiguity, because I thought that $\displaystyle f$ only named the relationship, and only stood for the fact that $\displaystyle f(x)$ and $\displaystyle x$ have a corresponding relationship. Also in function like$\displaystyle y = x + 1$, people say that "y is a function of x," but $\displaystyle f(x) = x + 1$, would you just say a function of x ($\displaystyle f(x)$) is a function of x?

May 10th, 2014, 07:45 AM  #2  
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108  Quote:
Quote:
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Note that there is some ambiguity between the function $f$ and the output $f(x)$ it produces on $x$. The function itself is often denoted by $f(x)$ where $x$ is a placeholder, not a fixed value. This ambiguity is resolved is some areas of mathematics where a function and its application to an argument are strictly distinguished.  
May 10th, 2014, 08:09 AM  #3 
Senior Member Joined: Feb 2014 From: Louisiana Posts: 156 Thanks: 6 Math Focus: algebra and the calculus  In that case, when one says "y is a function of x," are they talking about the process of adding 1, or the fact that the output, y, is dependent on the input, x? I think my lack of understanding comes to what the word "function" is denoting. I've seen it described as being the definition of when one variable quantity depends on another, but I have also seen it described as: in x + 1, the process of adding 1 is the function...

May 10th, 2014, 08:44 AM  #4 
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108 
If I understand correctly the two alternatives you are giving, they are as follows. (1) $y$ is a function of $x$ if there exists an process, an algorithm or rule for computing $y$ given $x$. (2) $y$ is a function of $x$ if $y$ completely depends on $x$ or is determined by $x$, i.e., once the value of $x$ is fixed, so is the value of $y$. Well, (1) is a special case of (2). If we know the rule for turning $x$ into $y$, then $y$ is definitely determined by $x$. For converse, one can try imagining a situation when $y$ is determined by $x$, but there is no way to come up with a formula or a way to find $y$ based on $x$. Whether this is possible is a philosophical question. One can argue that since $y$ is determined from $x$ "by the world", the rule that determines $y$ objectively exists even if we don't know it at the moment. One can also adopt a more subjective point of view where what is important is whether an algorithm for computing $y$ is known. There are subjects in math that try modeling both options. For example, the concept of function in the socalled constructive mathematics differs from the conventional one. Is this the distinction you have in mind? Usually a function is defined just as a set of inputoutput pairs. This corresponds to the objective view where the whole correspondence between the inputs and outputs exists as a single object. I don't think this is the only approach. I believe it is possible to think about a function as a rule mapping inputs to outputs leaving "rule" to be undefined similar to the word "set". 
May 10th, 2014, 10:03 AM  #5 
Senior Member Joined: Feb 2014 From: Louisiana Posts: 156 Thanks: 6 Math Focus: algebra and the calculus 
Thank you. I understand much better now. You answered my question perfectly.

May 10th, 2014, 10:12 AM  #6 
Senior Member Joined: Feb 2014 From: Louisiana Posts: 156 Thanks: 6 Math Focus: algebra and the calculus 
I have one last question though. You said that $\displaystyle f$ does represent the rule of correspondenceor, the algorithm for computing the output. If this is the case, then does $\displaystyle y$ in $\displaystyle y = x + 1$ represent that same thing that $\displaystyle f$ does in $\displaystyle f(x) = x + 1$? That is, does $\displaystyle y$ represent the process of adding $\displaystyle 1$ to $\displaystyle x$?

May 10th, 2014, 10:29 AM  #7  
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108  Quote:
 
May 10th, 2014, 11:02 AM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2664 Math Focus: Mainly analysis and algebra 
I would tend to read $y = x + 1$ as $y = f(x) = x+ 1$ meaning that $y$ is a variable that takes on values of the function $f$ which are determined by $x$ through the operation of addinng 1 to $x$.


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