0. Code

>	restart; with(plots):

Warning, the name changecoords has been redefined

>

SolidPlot := proc(f,a,b) local  box,i,n,x1,x2,xmid,delta,y1,y2,A,B, m,M,slope, concav: n:= 60;   delta := (b-a)/n;   x2 := a; for i from 1 to n do x1 := evalf(x2);          y1 := evalf( f(x1));   x2 := evalf(a + i*delta); y2 := evalf( f(x2));   B[i]:=polygonplot([[x1,0],[x1,y1],[x2,y2],[x2,0]], color=gold, style=patchnogrid); od; display({ seq( B[i],i=1..n )  } ); end:

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SolidPlotBound := proc(f,a,b, M) local  n,delta,x1,x2,y1,y2,B,i: n:= 60;   delta := (b-a)/n;   x2 := a; for i from 1 to n do x1 := evalf(x2);          y1 := evalf( f(x1)); x2 := evalf(a + i*delta); y2 := evalf( f(x2));   if(y1 > M) then y1 := M; else if (y1 < -M) then y1 := -M; fi;fi;   if(y2 > M) then y2 := M; else if (y2 < -M) then y2 := -M;fi;fi;   B[i]:=polygonplot([[x1,0],[x1,y1],[x2,y2],[x2,0]], color=red, style=patchnogrid); od; display({ seq( B[i],i=1..n )  } ); end:

 1.  The Inverse Sine Graph

The sine function is periodic, and therefore not one-to-one. And since only one-to-one functions have inverses, this makes a problem. Even the fundamental period is not one-to-one.

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display( [ plot(sin(x), x = -2*Pi..6*Pi, color = gold),          plot(sin(x), x = -0..2*Pi,    color = orange, thickness = 4),          polygonplot( [[0,-1],[2*Pi,-1],[2*Pi,1],[0,1]],          color =COLOR(RGB, 1, .8, .2), thickness = 0 ),          plot( [[0,-1],[2*Pi,-1],[2*Pi,1],[0,1]], color = red, linestyle = 2)] ); `Fundamental Period`;

However, we can find a piece of the sine which IS one-to-one, is continuous, and covers the full range of the sine function.

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display( plot(sin(x), x = -2*Pi..2*Pi, color = gold),          plot(sin(x), x = -Pi/2..Pi/2,    color = orange, thickness = 4),          polygonplot( [[-Pi/2,-1],[Pi/2,-1],[Pi/2,1],[-Pi/2,1],[-Pi/2,-1]],          color =COLOR(RGB, 1, .8, .2), thickness = 0 ),          plot( [[-Pi/2,-1],[Pi/2,-1],[Pi/2,1],[-Pi/2,1],[-Pi/2,-1]],                  color = orange, linestyle = 2) ); `Restricted Domain - which IS one-to-one`;

This is the graph of the restricted sine. Its domain is  and its range is [-1,1].

>	f := x-> sin(x); SolidPlot( f, -Pi/2, Pi/2);

The inverse sine is the mirror image of the restricted sine. Its domain is the range of the restricted sine, and its range is the domain of the restricted sine.

>	display( plot( arcsin(x), x = -Pi/2..Pi/2, color=blue, thickness=3, scaling=constrained),          plot( sin(x), x = -Pi/2..Pi/2, color=red, thickness=3, scaling=constrained));

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display( plot( arcsin(x), x = -Pi/2..Pi/2,                color = blue, thickness = 2, scaling = constrained),    SolidPlot( x-> sin(x), -Pi/2, Pi/2),    plot( [[-Pi/2,-1],[Pi/2,-1],[Pi/2,1],[-Pi/2,1],[-Pi/2,-1]],                  color = orange, linestyle = 2) ,    plot( [[-1,-Pi/2],[-1,Pi/2],[1,Pi/2],[1,-Pi/2],[-1,-Pi/2]],                  color = green, linestyle = 2),    plot( [[-Pi/2,-Pi/2],[Pi/2,Pi/2]], color = violet, linestyle = 3 ) );

 2.  The Inverse Cosine Graph

The cosine is also periodic, and therefore not one-to-one.

>	display( plot(cos(x), x = -2*Pi..6*Pi, color = blue),          plot(cos(x), x = -0..2*Pi,    color = red, thickness = 4),          plot( [[0,-1],[2*Pi,-1],[2*Pi,1],[0,1]], color = red, linestyle = 2) ); `Fundamental Period`;

As before, we can find a one-to-one piece of the cosine which covers the entire range of [-1,1]. However, the restriction is different than before

>	display( plot(cos(x), x = -2*Pi..6*Pi, color = blue),          plot(cos(x), x = 0..Pi,    color = red, thickness = 4),          plot( [[0,-1],[Pi,-1],[Pi,1],[0,1],[0,-1]],                  color = orange, linestyle = 2) ); `Restricted Domain - which IS one-to-one`;

This is the graph of the restricted cosine. Its domain is  and its range is [-1,1].

>	f := x-> cos(x); SolidPlot( f, 0, Pi);

The inverse cosine is the mirror image of the restricted cosine. Its domain is the range of the restricted cosine, and its range is the domain of the restricted cosine.

>	display( plot( arccos(x), x = -1..Pi, color=blue, thickness=3, scaling=constrained),    SolidPlot( x-> cos(x), 0, Pi)  );

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display(    plot( arccos(x), x = -1..Pi, color=blue, thickness=3, scaling=constrained),    SolidPlot( x-> cos(x), 0, Pi),    plot( [[0,-1],[Pi,-1],[Pi,1],[0,1],[0,-1]],                  color = orange, linestyle = 2) ,    plot( [[-1,0],[-1,Pi],[1,Pi],[1,0],[-1,0]],                  color = green, linestyle = 2),    plot( [[-Pi/2,-Pi/2],[Pi,Pi]], color = violet, linestyle = 3 ) );

 3.  The Inverse Tangent Graph

The tangent, too, is periodic.

>	M := 15: display( plot(tan(x), x = -2*Pi..3*Pi, y = -M..M,color = blue, discont = true),          plot(tan(x), x = -Pi/2..Pi/2, y = -M..M,color = red, thickness = 4, discont = true),          plot( [[-Pi/2,-M],[Pi/2,-M],[Pi/2,M],[-Pi/2,M],[-Pi/2,-M]], color = red, linestyle = 2) );

The fundamental period of the tangent IS one-to-one (except at the places it's undefined of course). Therefore, we can restrict the tangent to the open interval  to get a restricted version of the tangent which is one-to-one,continous, and covers the complete range of all real numbers.

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M := 15: display( plot(tan(x), x = -Pi..2*Pi, y = -M..M,color = blue, discont = true),          plot(tan(x), x = -Pi/2..Pi/2, y = -M..M,color = red, thickness = 4, discont = true),          plot( [[-Pi/2,-M],[Pi/2,-M],[Pi/2,M],[-Pi/2,M],[-Pi/2,-M]], color = red, linestyle = 2) ); `Restricted Domain - which IS one-to-one`;

This is the graph of the restricted tangent. Its domain is [-Pi/2,Pi/2] and its range is [-1,1].

>	SolidPlotBound( x->tan(x), -Pi/2+.1, Pi/2-.1, M);

The inverse tangent is the mirror image of the restricted sine. Its domain is the range of the restricted sine, and its range is the domain of the restricted sine.

>	M := 10: display( plot( arctan(x), x = -M..M, color=blue, thickness=3, scaling=constrained),   SolidPlotBound( x->tan(x), -Pi/2+.03, Pi/2-.03, M)  );

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display(    plot( arctan(x), x = -M..M, color=blue, thickness=3, scaling=constrained),    SolidPlotBound( x->tan(x), -Pi/2+.03, Pi/2-.03, M),    plot( [[-Pi/2,-M],[Pi/2,-M],[Pi/2,M],[-Pi/2,M],[-Pi/2,-M]],             color = red, linestyle = 2), plot( [[-M,-Pi/2],[-M,Pi/2],[M,Pi/2],[M,-Pi/2],[-M,-Pi/2]],color = blue, linestyle = 2),    plot( [[-M,-M],[M,M]], color = violet, linestyle = 3 )   );

Notice that this variation of the tangent maps the real line into the open interval (0,1) as a one-to-one function - showing that there are same number of real numbers as there are real numbers in between 0 and 1.
 

>	f := x -> ((2/Pi)*arctan(x) + 1)/2;

>	plot( f(x), x = -30..30);

>