Finding the graph of functions How do I find the graph of the below functions?


*

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

*$g_n(x)=x^n-x^{n-1}$, where $n\in \mathbb{N}$
Help greatly appreciated! 
 A: 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 $n=0,\ldots,10$:
                               
and here is a plot of the graphs of $g_n$ for $n=1,\ldots,11$:
                               
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.)
