The Journal of Online Mathematics and Its Applications, Volume 7 (2007)

GeoGebra, Markus Hohenwarter and Judith Preiner

The following construction protocol explains how to draw the graph of the composition of two functions `g`(`x`) and `h`(`x`).

Use the buttons in the navigation bar to go through our construction step by step.

- We create the functions
`g`(`x`) = sin(3`x`) and`h`(`x`) =`x`^{2}/ 4 by using the input field. This gives us the equations of the functions in the algebra window and their graphs in the geometry window. - Then we draw the median line
`m`:`y`=`x`and place a point`x`_{0}on the`x`-axis. - To get the graph of the composite function
`f`(`x`) =`g`(`h`(`x`)) we first draw a vertical segment from`X`_{0}to the inner function`h`(`x`) which gives us point`A`. - Then we continue horizontally towards the median
`m`:`y`=`x`to get point`B`. - Afterwards we proceed vertically towards the outer function
`g`(`x`) where we reach point`C`. - By closing the rectangle at point
`D`with the last horizontal segment, we get the value`f`(`x`_{0}) above or below our starting point`x`_{0}.

Point `D` has the `x`-coordinate `x`_{0} while its `y`-coordinate is `f`(`x`_{0}). By dragging point `x`_{0} with the mouse you can examine the course of the composite function `f`(`x`) = `g`(`h`(`x`)) which is created as the trace of point `D` (see Haftendorn, 2005, p. 235). To turn the trace on, right click (Mac OS: apple click) on point `D` and check "Trace on".

We did the construction of the vertices `A`, `B`, `C` and `D` by intersecting lines and graphs of functions. Another way would be to calculate the points directly using the input field, e.g. `A = (x(X_0), h(x(X_0))) or `

`B = (y(A), y(A)).`

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