$What is the graph of $f(y)=cy$?$

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$What is the graph of $f(y)=cy$?$

Graphs of Functions

Webtolight

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The graph of the function is a straigh line passign through the origion (see the attached file).

The set deifnition of the graph of a function $f$ is given by
\[G_f=\{(x,y)\in \mathbb{R}\times \mathbb{R}: y=f(x) \}=\{(x,f(x)): x \in \mathbb{R}\}.\]

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Note that
\[\{(x,f(x)): x \in \mathbb{R}\}=\{(y,f(y)): y \in \mathbb{R}\}\]
Indeed $x$ and $y$ are just the name of the variables, and it does not matter if we use $x$ or $y$.
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In the graph of a function, the vertical axis is typically labeled as $f(y)$ (or more commonly $f(x)$) to indicate that it represents the outputs of the function $f$ for each input value $y$ along the horizontal axis. The horizontal axis is labeled as $y$ (or more typically $x$), which represents the independent variable, the input to the function. Labeling the vertical axis as $f(y)$ clearly shows that the values on this axis are the results of applying the function $f$ to the values of $y$ in the horizental axis. This labeling helps clarify the relationship between the input and the output in the function.

Yes, we can definitely change the variables used for the horizontal and vertical axes to any variables you want when graphing a function. In mathematics, the choice of symbols to represent variables is largely arbitrary and can be chosen to best suit the context or to adhere to specific conventions in certain fields.

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Erdos

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Erdos

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Let me know if you need any clarifications.
Webtolight

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Erdos, I appreciate the response. So in your example, did you set y=f(y). Or you just labeled the horizontal axis as f(y). In the set definition, wouldn't it be (y, f(y): y is in R) for the pairs?
- Erdos
  
  0
  
  I added some more details.
Webtolight

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Great! One last thing . Suppose that the input is now x+y=u. Is this also valid: Graph of f = {(u,v) ∈ R×R : v = f(u)} is equal to {(x,y)∈R×R:y=f(x)}.
- Erdos
  
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  Your question does not make much sense. In the equation x+y=u there are three variables. You should have two variables in the equation of the graph of a function.
Webtolight

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I am sorry about that. I hope I can clarify. What if we add two real values, x and y. Is it true that the addition of two real numbers = another real number. Then apply the function to that new variable which is composed of the addition of two real variables? Can we graph this new variable?
Webtolight

0

For example: f(x+y)=c(x+y).
- Erdos
  
  +1
  
  In this case you can set z=x+y, and the equation can still be written as f(z)=cz. The graph will be the same. If you define f(x,y)=c(x+y) then this is a function of two variables x and y whose graph is a surface in three dimensions, but I think this is outside of the scope of your question.
Webtolight

+1

Yes it would be. So then we would set up the graph to be the vertical axis as v and the horizontal axis as u. Therefore this statement would be true: Graph of f = {(u,v) ∈ R×R : v = f(u)}. I should have set that expression equal to another variable as there are only two variables in the equation of the graph of a function. I understand now.

$What is the graph of $f(y)=cy$?$

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