0. Code

>	restart; with(plots):

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1. Functions & Graphs

Functions can be expressed as graphs too. For each value of x, there is a value of f(x). If we let y = f(x), we have a point (x,y). If we graph all of these points, we have a "picture" of what the function looks like. The graph of a function tells us many thing about the relationship between x and y.

>	f := x -> x^3 - 20*x^2 + 100*x - 40; plot( f(x), x = -3..18, y = -120..120, thickness = 2, color = green);

Functions can take many different forms. Some of these functions are beyond the scope of your algebra class, but you can just back and enjoy the scenery for a moment. You don't have to be a chef to enjoy a great meal.

>	f := x -> sqrt( abs(sin(x) + cos(x) )): plot( f(x), x = 0..4*Pi, y = 0..1.3, thickness = 2, color = gold);

>	f := x -> exp( x - 3*sqrt(x)*sin(x) ) - exp( x + 3*sqrt(x)*cos(x) ); plot( f(x), x = 0..3,  thickness = 2, color = magenta);

>	f := x -> floor(1 + sin(8*x)) * x; plot( f(x), x = 0..10,  thickness = 2, color = aquamarine, discont = true);

>	f := x -> sin( exp(1/cos(x)) ); plot( f(x), x = 0..3,  thickness = 1, color = green);

>	restart: with(plots):

Warning, the name changecoords has been redefined

>	f := x -> floor( 5*tan(x) ); plot( f(x), x = 0..2*Pi, y = -10..10,              thickness = 3, color = tan, discont = true);

>	f := x -> sqrt( x + abs(sin(x) - cos(x) )); plot( f(x), x = 0..10*Pi, y = 0..6, thickness = 2, color = blue);

>	f := x -> exp( x/3 ) + x*(1 + sin(x))*cos(x^1.5); plot( f(x), x = 0..10,  thickness = 2, color = sienna);

>	f := x -> abs(x) - abs(x-3) + abs(x-5) - abs(x-9); plot( f(x), x = -5..14,  thickness = 2, color = orange, discont = true);

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2. Common Functions

Lets examine some of the garden variety functions we'll find sprouting up in the text.

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restart:

>	f := x -> 3*x - 4;  plot( f(x), x = -3..3,                         title="(non vertical) lines are functions too!");

>	f := x -> x^2;  plot( f(x), x = -3..3, title="Classic Parabola");

>	f := x -> -x^2;  plot( f(x), x = -3..3, title="Classic Inverted Parabola");

>	f := x -> 12-(x-3)^2;  plot( f(x), x = -2..7, title="Another Parabola");

>	f := x -> abs(x);  plot( f(x), x = -3..3, title="Classic Absolute Value");

>	f := x -> sqrt(x);  plot( f(x), x = -1..16, title="Squareroot");

>	f := x -> x^3;  plot( f(x), x = -3..3,y = -15..15, title="Classic Cubic");

>	f := x -> x^3 - 4*x;  plot( f(x), x = -3..3,y = -10..10, title="Curvy Cubic");

>	f := x -> 1/x;  plot( f(x), x = -3..3, y = -10..10,                      title="Reciprocal", discont = true);

3. Vertical Line Test

When looking at a graph it is fairly easy to see if its the graph of a function or not.

Vertical Line Test - (tests for a function)

If EVERY vertical line crosses a graph in at most one point, then the graph is the graph of a function. Here are some examples of graphs which FAILS this test. Thus none of these are functions. They are merely relations.

>	restart: with(plots):

Warning, the name changecoords has been redefined

>	expr := 4*x^2 - 3*y^2 + y^3 = 1; plots[display](      implicitplot( expr ,x=-2..2, y=-2..4,color = blue, thickness = 2),      plot( {seq([[k/4,-2],[k/4,4]], k=-8..8)}, x=-2..2, y= -2..4,                  color = gold));

>	expr := y^2 = x + 1.5; plots[display](      implicitplot( expr ,x=-2..2, y=-2..2,color = blue, thickness = 2),      plot( {seq([[k/4,-2],[k/4,2]], k=-8..8)}, x=-2..2, y= -2..2, color = gold));

>	restart: with(plots):

Warning, the name changecoords has been redefined

>	expr := 3*(x)^2 + 9*(y)^4 = 8; plots[display](      implicitplot( expr ,x=-2..2, y=-1..1,                  color = blue, thickness = 2, numpoints = 1000),      plot( {seq([[k/4,-1],[k/4,1]], k=-8..8)}, x=-2..2, y= -1..1, color = gold));

>

Here are some examples which PASS the Vertical line test - thus they are functions.

>	restart: with(plots):

Warning, the name changecoords has been redefined

>	f := x -> x^2; plots[display](      plot( f(x) ,x=-2..2, y=-2..4,color = green, thickness = 2),      plot( {seq([[k/4,-2],[k/4,4]], k=-8..8)}, x=-2..2, y= -2..4,            color = tan));

>	f := x -> 2+2*sin(Pi*x); plots[display](      plot( f(x) ,x=-2..2, y=0..4,color = green, thickness = 2),      plot( {seq([[k/4,-2],[k/4,4]], k=-8..8)}, x=-2..2, y= 0..4,            color = tan));

>	f := x -> 1 + abs(x+1)-abs(x-1); plots[display](      plot( f(x) ,x=-2..2, y=-2..4,color = green, thickness = 2),      plot( {seq([[k/4,-2],[k/4,4]], k=-8..8)}, x=-2..2, y= -2..4,            color = tan));

>	f := x -> 1/x; plots[display](      plot( f(x) ,x=-2..2, y=-5..5,color = green, thickness = 2, discont = true),      plot( {seq([[k/4,-5],[k/4,5]], k=-8..8)}, x=-2..2, y= -5..5,            color = tan));

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