Whew! That got a little confusing. Let's back up for a second and just talk about functions, and ignore Geogebra for a moment.[br][br]Let's write out in plain English what each our our functions from the previous activity actually [i]do[/i]:[br][br][list=1][*][code]f[/code] -- multiplies its input by 2[/*][*][code]g[/code] --adds 2 to its input[/*][/list][br]We can [b]compose[/b] these functions by running one first, and the the other second. There's, or course, two ways we could do this:[br][br][list=1][*]We could run [code]f[/code] first and then we could run g on the outputs of [code]f[/code], or[/*][*]We could run [code]g[/code] first and then we could run f on the outputs of [code]g[/code][/*][/list][br]Let's see what each of these do to an input of x equal to 1[br][list=1][*][code]f[/code] first, then [code]g[/code]: 1 times 2 is 2; 2 plus 2 is 4[/*][*][code]g[/code] first, then [code]f[/code]: 1 plus 2 is 3; 3 times 2 is 6[/*][/list][br]Now you can see why the two are different![br][br]Also, just to be clear, combining functions in this way (first one, then another) [i]is[/i][b] function composition. [/b][br][br]The notation that mathematicians (and Geogebra!) use for function composition is:[br][br][list=1][*][code]f[/code] first, then [code]g[/code]: [code]g(f(x))[/code][/*][*][code]g[/code] first, then [code]f[/code]: [code]f(g(x))[/code][/*][/list][br]If this seems backwards to you, the reason for the notation is that you start with the input, x, and then work your way out from there. [br][br]Click through to the next activity for another example to help crystalize this concept.