Construction of Inverse Function
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User Guide
Construction of Inverse Function of 𝒇(𝒙) = 𝒔𝒆𝒄(𝒙)
GeoGebra Applet: Inv_secx.ggb
Link: http://geogebrawiki.wikispaces.com/Inverse+function_sec(x)
GeoGebra tools:
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Insert text: Creates static and dynamic text or LaTeX formulas in the Graphics
View.
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Move: Drag and drop free objects with the mouse.
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Slider: Creates a slider for a number or an angle.
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Perpendicular Line: Selecting a line g and a point A creates a straight line
through A perpendicular to line g.
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Intersect two objects: Creates point as intersection of two objects.
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Point: Creates new point in the drawing pad in the Graphics window.
GeoGebra commands:
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1.
Reflect[Object,Line]: Reflects Object about Line.
Extremum[Polynomial]: Yields all local extrema of the polynomial function as points
on the function graph.
Asymptote[Function]: Yields both asymptotes of the Function.
Element[List, Number n]: Yelds the nth element of the list.
Intersect[Object, Object]: Yields intersection point od two objects.
Function[f, Number a, Number b]: Yields a function graph, that is equal to f on the
interval [a, b] and not defined outside of [a, b].
Click on the
Insert text tool and then in Graphics window to insert text. A dialog window
opens (Fig.1). In the Edit window insert text Graph of inverse function of f(x)=sec(x)=1/cosx.
Click Ok. The text is inserted in the Graphics window.
Use the
Move tool to change the position of the text in the Graphics window. Click on the
tool then on the text and drag it to new position.
To change the color of the text right click on the text and choose
on the Color tab and choose the color.
Object Properties. Click
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Construction of Inverse Function
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Fig.2
Fig.1
2.
Create slider to make drawing of different curves in the Graphics window step by step as they
are constructed. Click on the
Slider tool and then in Graphics window. A dialog window
opens (Fig.2). GeoGebra suggests a for name of the slider (you can choose another name). Insert
1 for min, 8 for max and 1 for Increment. Now slider can get value that is one of the numbers
from 1 to 8. Moving the point on the slider will cause drawing of different curves one by one in
order to give better visualisation of the concept of function – inverse function (this will be set
later in the construction).
Click on the Slider tab and choose Vertical to make the slider position vertical in the Graphics
window (default option is horizontal). Uncheck the button
to make the slider moveable
on the screen. Insert 140 for the width of the slider.
Click on the Animation tab to change the speed of moving the point on the slider (default is 1)
and the way of moving the point on the slider (Oscillating, Increasing, Decreasing and
Increrasing (Once)). Choose Oscillating to move the point on the slider up and down.
3.
Insert text Show curve, Show Line y=x, Show reflected curve, Show asymptotes, Show
extremum of the curve on arbitrary interval, Restrict original curve to make the reflected curve
a function , Check inversity (slide point A), change the color of each of them and put them next
to the slider a (Fig.3).
Fig.3
4.
Now draw the graph of 𝒇(𝒙) = 𝒔𝒆𝒄(𝒙).
Type f(x)=1/cos(x) in the Input bar in the down left bottom of the GeoGebra window and press
the Enter key. The curve is drawn in Graphics window and equation of the curve is written in the
Algebra window (Fig.4). To change the look of the curve right click on the equation of the
curve in the Algebra window or graph in the Graphics window and choose Options. Click on
the Color tab to change the color and the Style tab to change style and thickness of the curve.
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Construction of Inverse Function
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Fig.4
Fig.5
5.
Type y=x in the Input bar to draw the line of the reflection and press the Enter key. The line is
drawn in the GeoGebra window. Look in the Algebra window. GeoGebra names the line b:y=x.
(a is already reserved for the the name of the slider).
6.
Use command Reflect[ <Object>, <Line>] to reflect Object about Line.
Start typing g(x)=Reflect[f(x),b] in the Input bar to reflect the curve of the function f(x) about
line of reflection b:y=x. If GeoGebra suggests the desired command, hit the Enter key in order
to place the cursor within the brackets. If the suggested command is not the one you wanted to
enter, just keep typing until the suggestion matches.
The reflected curve is drawn in the Graphics Window and equation of the curve is written in the
Algebra window (Fig.5).
7.
Make the Vertical line test to determine whether a graph is the graph of a function and the
Horizontal line test to determine if a function has an inverse that is also a function.
How to use vertical line test? Ask: Is it possible to draw a vertical line that intersects the graph in
two or more places? If so, then the graph is not the graph of a function. If it is not possible,
then the graph is the graph of a function.
How to draw vertical line? Use the
Perpendicular Line tool to draw line that passes
through arbitrary point and is normal to defined line. Click on the tool, then in the Graphics
window to mark arbitrary point and then click on x – axis (defined line).There is only one
intersection point of the line and the graph of the curve (Fig.6). The result will be the same, one
intersection point of a line and the graph of the curve, for every drawn arbitrary vertical line. So
the graph is a graph of a function.
The horizontal line test is used to determine if a function has an inverse that is also a function.
We seek to answer this question: is it possible to draw a horizontal line that intersects the graph
in two or more places? If so, then the graph is not the graph of a function whose inverse is also a
function. We also say that if a graph passes the horizontal line test – in other words, if it is not
possible to draw a horizontal line that intersects the graph in two ore more places – then the
graph is one-to-one, another way of saying that its inverse is a function.
Now use the same tool to draw arbitrary horizontal line. There are more than one intersection
points of the line with the graph of function f(x) (Fig.7). That means that the inverse (reflected)
graph is not graph of a function.
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Construction of Inverse Function
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Fig.6
8.
Fig.7
We must restrict the original function f(x) to make the reflected curve also a function. First we'll
find asymptotes of the function and then we'll restrict the function in the interval between the
asymptotes.
Command Asymptote[Function] yelds both asymptotes of a function as elements of list.
In the Input bar type ListAsym=Asymptote[f] to find asymptotes of function f(x) and press the
Enter key. The asymptotes are drawn in the Graphics window and list ListAsym, with elements
asymptotes od the function, is created in the Algebra window (Fig.8).
We'll find the left border as intersection point of asymptote x=-1.57 (the second element of list
ListAsym) and x-coordinate axis (y=0). The right border is intersection point of asymptote
x=1.57 (the first element of list ListAsym) and x-coordinate axis (y=0).
Command Element[List, Number n] yields the nth element of the list.
Command Intersect[Object, Object] yields intersection point od two objects (lines).
Type LeftBorder=Intersect[Element[ListAsym, 2], y = 0] in the Input bar and press the Enter
key. Type RightBorder=Intersect[Element[ListAsym, 1], y = 0] in the Input bar and press the
Enter key. The coordinates of the points are written in the Algebra window and the points are
marked in the Graphics window (Fig.9).
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Construction of Inverse Function
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Fig.9
Fig.8
Now find extremum of the curve in the interval between LeftBorder and RightBorder.
Use command Extremum[ <Function>, <Start x-Value>, <End x-Value> ] to find Extremum of
Function in interval [Start x-Value, End x –Value].
Type Extf=Extremum[f, x(LeftBorder), x(RightBorder)] in the input bar and press the Enter key.
The coordinates of the point Extrf are written in the Algebra window and the point is marked in
the Graphics window (Fig.10).
Fig.10
Use the command Function[ <Function>, <Start x-Value>, <End x-Value> ] to restrict a defined
function in interval [Start x-Value, End x-Value].
Write f1(x)= Function[f, x(Extrf), x(RightBorder)] in the Input bar and press the Enter key. Now
function f(x) is restricted on interval [x(Extrf),x(RightBorder)], where x(Extrf) is the x
coordinate of the point Extrf and x(RightBorder) is the x coordinate of the point RightBorder.
To reflect f1(x) about line b:y=x, insert g1(x)= Reflect[f1, b] in the Input bar and press the Enter
key.Now f1(x) passes both vertical and horizontal line test, so the inverse g1(x) is also a function
(Fig.11).
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Construction of Inverse Function
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Fig.11
9.
Mark point to check inversity. Click on the
f1(x) to mark point A.
Point tool and on the graph of the function
Functions f(x) and g(x) are inverse of one another if f(g(x)) = x and g(f(x))=x, for all values of x
in there respective domains. So if point A has coordinates (x,y) and lies on graph of the function
f(x), then y=f(x). Then g(f(x))=g(y)=x must be true. That means that point B with coordinates
(y,x) lies on the graph of the function g(x).
Type B= (y(A), x(A)) in the Input bar, where y(A) means y coordinate of point A, and x(A)
means x coordinate of point A and press the Enter key. See that pont B lies on the curve of the
function g1(x) (Fig.12). Moving the point A through the graph of function f1(x) indicates
moving of the point B through the graph of the function g1(x). That means that if A(x,y) is a
point that lies on the graph of function f(x), there is only one point B(y,x) that lies on the graph
of a function g(x).
Fig.12
10. Now make the drawing of the constructed curves dependent of a condition fulfilled. This allows
drawing of the curves step by step by moving the point on the slider down and up.
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Construction of Inverse Function
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Right click on the graph of the function f(x) in the Graphics window and choose
Object
Properties. Click on the Advanced tab and type a < 8 ∧ a > 2 in Condition to Show Object.
The graph of the function f(x) will be drawn in the Graphics window everytime the value of the
slider a is between 3 and 7.
Repeat the procedure to make conditional appeareance of the line b (insert a<7), the graph of
g(x) (insert a<6 ∧ a>2), asymptotes of f(x) (insert a<5 ∧ a>2 ), points LeftBorder and
RightBorder (insert a=3 ∨ a = 4), point Extrf (insert a=3), the graph of f1(x) (insert a=1 ∨
a=2), the graph of g1(x) (insert a=1 ∨ a=2), and the points A and B (insert a=1).
11. Testing the construction:
Move the point on the slider down or up to make drawing of the curves one by one.
Move the point A to check inversity.
12. View construction protocol:
From the View menu choose Construction protocol to view protocol of the construction.
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