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May 10th, 2014, 07:18 AM   #1
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Question 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 correspondence--that 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?
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May 10th, 2014, 07:45 AM   #2
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Does $\displaystyle f$ represent the rule of correspondence--that is, multiplying by 2 and adding 1
Yes.

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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.
Why does saying that an input $x$ goes through $f$ to produce an output $f(x)$ not square with saying that $f$ is the relationship between the input and the output? To me they say almost the same thing.

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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?
I am not sure I understand the question. The equation $\displaystyle y = x + 1$ says that $y$ is a function of $x$, and it can be denoted by $y(x)$.

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.
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May 10th, 2014, 08:09 AM   #3
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The equation $\displaystyle y = x + 1$ says that $y$ is a function of $x$, and it can be denoted by $y(x)$.
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...
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May 10th, 2014, 08:44 AM   #4
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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 so-called 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 input-output 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".
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May 10th, 2014, 10:03 AM   #5
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Thank you. I understand much better now. You answered my question perfectly.
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May 10th, 2014, 10:12 AM   #6
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I have one last question though. You said that $\displaystyle f$ does represent the rule of correspondence--or, 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 $\displaystyle x$?
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May 10th, 2014, 10:29 AM   #7
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I have one last question though. You said that $\displaystyle f$ does represent the rule of correspondence--or, 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 $\displaystyle f(x) = x + 1$?
Can you explain what difference you see between $\displaystyle y = x + 1$ and $\displaystyle f(x) = x + 1$? Are you saying it is important that there is an $(x)$ following $f$ or that $y$ is the name of a variable vs the name of a function? Why do you see these equalities as different?
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May 10th, 2014, 11:02 AM   #8
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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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