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How do I find the graph of the below functions?

  1. $f_n(x)=x^n$ where $n\in \mathbb{N}\cup\{0 \}$

  2. $g_n(x)=x^n-x^{n-1}$, where $n\in \mathbb{N}$

Help greatly appreciated!

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You need to explain what you mean by "graph". Do you want a picture? Or the set $$\{(x,y)\in (\text{domain})\times(\text{codomain})\mid y=f(x)\}\quad ?$$ –  Zev Chonoles Apr 8 '13 at 7:44
    
Yes, like by sets or description. –  user67253 Apr 8 '13 at 7:45
    
What are the domains and codomains of the functions $f_n$ and $g_n$? The real numbers $\mathbb{R}$? The complex numbers $\mathbb{C}$? –  Zev Chonoles Apr 8 '13 at 7:47
    
It's described after "where" –  user67253 Apr 8 '13 at 7:56
    
As far as I can tell, $n$ is just an index, and the question is asking you to find the graph of any one of the functions $f_n$ and $g_n$. Or are you claiming the question is considering $f$ and $g$ as functions from $\mathbb{N}$ to (the set of functions)? Even if the latter is the case, we still need to know what the domains and codomains of $f_n$ and $g_n$ are. Is $f$ a function from $\mathbb{N}\cup\{0\}$ to (the set of functions $\mathbb{R}\to\mathbb{R}$)? or to (the set of functions $\mathbb{N}\to\mathbb{C}$)? etc. –  Zev Chonoles Apr 8 '13 at 8:02
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1 Answer

up vote 0 down vote accepted

If the question is interpreted as asking:

For any given $n\in\mathbb{N}\cup\{0\}$, what is the range of $f_n:\mathbb{R}\to\mathbb{R}$, defined by $f_n(x)=x^n$?

For any given $n\in\mathbb{N}$, what is the range of $g_n:\mathbb{R}\to\mathbb{R}$, defined by $g_n(x)=x^n-x^{n-1}$?

Then for any $n\in\mathbb{N}\cup\{0\}$, we have $$\mathrm{range}(f_n)=\{(x,x^n)\in\mathbb{R}^2\mid x\in\mathbb{R}\}$$ and for any $n\in\mathbb{N}$, we have $$\mathrm{range}(g_n)=\{(x,x^n-x^{n-1})\in\mathbb{R}^2\mid x\in\mathbb{R}\}$$


If the question is interpreted as asking:

What is the range of the map $F:\mathbb{N}\cup\{0\}\to\mathbb{R}^\mathbb{R}$, defined by $F(n)=f_n$ where $f_n(x)=x^n$?

What is the range of the map $g:\mathbb{N}\to\mathbb{R}^\mathbb{R}$, defined by $G(n)=g_n$ where $g_n(x)=x^n-x^{n-1}$?

Then $$\mathrm{range}(F)=\{(n,f_n)\in(\mathbb{N}\cup\{0\})\times\mathbb{R}^\mathbb{R}\mid n\in\mathbb{N}\cup\{0\} \}$$ and $$\mathrm{range}(G)=\{(n,g_n)\in\mathbb{N}\times\mathbb{R}^\mathbb{R}\mid n\in\mathbb{N}\}.$$


If the question is interpreted as asking for a depiction of the graphs of $f_n$ and $g_n$, then here is a plot of the graphs of $f_n$ for $n=0,\ldots,10$:

                               enter image description here

and here is a plot of the graphs of $g_n$ for $n=1,\ldots,11$:

                               enter image description here

Mathematica code:

max = 10

listofplots = Table[Plot[Evaluate@Table[x^m, {m, 0, n}], {x, -2, 2}, 
  PlotRange -> {-5, 5}, AspectRatio -> 1, PlotStyle -> 
  Table[Directive[Hue[t/(max + 1)], Thick], {t, 0, n}]], {n, 0, max}]

listofplots2 = Table[Plot[Evaluate@Table[x^m - x^(m - 1), {m, 1, n}], {x, -2, 2},
  PlotRange -> {-5, 5}, AspectRatio -> 1, PlotStyle ->
  Table[Directive[Hue[t/(max + 1)], Thick], {t, 0, n - 1}]], {n, 1, max + 1}]

Export["animation.gif", listofplots, "DisplayDurations" -> {1}]

Export["animation2.gif", listofplots2, "DisplayDurations" -> {1}]

(I know, this is rather anticlimactic, but there really isn't anything else to say.)

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