Graphical Relationship Between Inverse Functions

The graphs  of a function and  its inverse are reflections about the line y=x. That is, if (x,y) is a point on the graph of the function  then the point (y,x) is a point on the graph of the inverse. The transform command from Maple's plottools package can be used to implement this graphical relationship between a function and its inverse:

>	INV := plottools[transform]( (x,y) -> [y,x] ):

This can be used as follows:

>	display( INV( P_tan4 ), view=[-10..10,-10..10] );

>

Note that this graph cannot be the graph of a function. The problem is that each value of x  has (at least) four distinct values. The usual way to eliminate this problem is to attempt to reduce the  domain of the original function so that each x has exactly one y value. There are many ways to do this for the tangent (or any periodic) function. The standard domain for the tangent function is [-Pi/2,Pi/2].

>	P_tan := plot( tan(x), x=-Pi/2..Pi/2, y=-10..10, discont=true, scaling=constrained ): P_tan;

>	display( INV( P_tan ), view=[-10..10,-10..10] );

It is very simple to plot the function and its inverse in the same graph.

>	display( [P_tan,INV(P_tan)], view=[-10..10,-10..10] );

However, it is not so simple to combine these plots with different colors or other distinguishing marks. The following Maple procedure plots the original function, its inverse (in blue), and the line y=x. (Remember to execute this command but do not make any changes to it!)

>	plot_reflect := proc(P)   local opts, Paxis;   if nargs>1     then opts := args[2..-1]     else opts := NULL   end if;   Paxis := plot( x, x=-10..10, color=black ):   plots[display]( [P, Paxis, subsop([1,2]=COLOUR(RGB,0.0,0.0,1.0), INV( P ) )], opts ): end proc:

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To illustrate its use, here is a single graph showing the tangent function on the interval [-Pi/2,Pi/2],  its inverse, and the line y=x. Note how the blue curve is the mirror image of the red curve across the (black) line y=x.

>	plot_reflect( P_tan, title="The Tangent Function and Its Inverse", view=[-10..10,-10..10] );

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Graphical Relationship Between Derivatives of Inverse Functions

The relationship between the derivative of a function and the derivative of its inverse is also easy to visualize. Recall that the tangent line to a curve at a point is the limit of a set of secant lines.

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plot_secantline := proc(f,domain::name=range,x1::realcons,x2::realcons)   local a, b, opt, Pf, Plx, Ply, Psl, Ptr, x, X, Y, y1, y2;   if not typematch(domain,x::name=a::realcons..b::realcons)     then error("invalid arguments")   end if;   if nargs>4     then opt := args[5..-1]     else opt := NULL   end if;   Pf := plot( f, domain, color=red );   y1 := eval(f,x=x1);   y2 := eval(f,x=x2);   X := x1 + (x2-x1)*t;   Y := y1 + (y2-y1)*t;   Psl := plot( [ X, Y, t=-3..3 ], color=pink );   Ptr := plot( [ [x1,y1], [x2,y1], [x2,y2] ], color=cyan );   Plx := plots[textplot]( [ (x1+x2)/2, y1, "dx" ], align=BELOW );   Ply := plots[textplot]( [ x2, (y1+y2)/2, "dy" ], align=RIGHT );   plots[display]( [Pf, Psl, Ptr, Plx, Ply], opt ); end proc:

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For example, here is the graph of the tangent function with the secant line through the points on y=tan(x) with x=1.3 and x=1.4.

>	P_tansec := plot_secantline( tan(x), x=-Pi/2..Pi/2, 1.3, 1.4, view=[-10..10,-10..10] ): P_tansec;

This secant line is very steep -- and hence the value of the derivative will be very large (and positive).

The following command produces a nice picture of the secant line for the function and the corresponding secant line for the inverse function.

>	plot_reflect( P_tansec, view=[0..6,0..6] );

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Maple and "Functions"

Note the different way in which this definition is made. The arrow operator (->) is used to define a function when you know the explicit rule used to define the function.

The benefit of this form is that we do not need to use eval to evaluate the function at a point. For example, the value at x=3 can be found with

>

f(3);

A little later in this worksheet the unapply  command will be used to create another function. This method must be used when you want to create a function from a Maple result.

Regardless of how the function is defined, it is easy to evaluate a function at a specific point. For example,

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f(3);

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f(a+b);

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Points to Ponder

• How would you compute this derivative by hand?
• Which form  - if either - looks more like what you would expect to find?

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Additional Note about Maple's solve  Command

Note that this result could be found by asking Maple to solve the inequality

>	need_to_show;

for x as follows:

>	solve( need_to_show, x );

While this works for this example, do not be surprised if Maple is unable to return such a useful result for other examples that you might consider.

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Points to Ponder

• Are you concerned that the graphs of the function and its inverse intersect?

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Step 1

Solve the equation y=f(x) for x in terms of y.

First, setup the equation to be solved

>	q0 := y = f(x);

then solve for x in terms of y

>	q1 := solve( q0, {x} );

we see that there are two solutions. These two solutions arise from the solution of a quadratic equation. Which one -- if either -- is appropriate as the inverse function for this example? To test, let's plot both functions:

>	x1 := eval( x, q1[1] );

>	x2 := eval( x, q1[2] );

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>	plot( [x1,x2], y=-6..10, color=[blue,green] );

Aha! Only the blue curve is the reflection of the original function.

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Step 2

Use f^(-1)(y) to name the resulting expression in y.

>	f_inv := unapply( x1, y );

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Step 3

Replace y by x to get the formula for f^(-1)(x).

Note that the way in which Step 2 has been completed in Maple means this step is no longer necessary.

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f_inv(x);

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