GeoGebra

Composition of Functions Part 3

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

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