Unit 1A

Precalculus

Lesson 3: Compositions of Functions

In the previous Lesson, you added, subtracted, multiplied, and divided functions. It is also possible to create a new function by embedding one, or more, in another. This process is known as the composition of functions.

During the BP Deepwater Horizon oil spill, the area of the oil spill could be modelled as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»r«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»t«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»7«/mn»«mi»t«/mi»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»t«/mi»«/mstyle»«/math» is the time in days and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»r«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is the radius of the oil spill in kilometres. The area of the spill could be modelled using «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»A«/mi»«mfenced»«mi»r«/mi»«/mfenced»«mo»=«/mo»«mo»§#960;«/mo»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math». It is possible to apply the radius function, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»r«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»t«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»7«/mn»«mi»t«/mi»«/mrow»«/mstyle»«/math», to the area function, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»A«/mi»«mfenced»«mi»r«/mi»«/mfenced»«mo»=«/mo»«mo»§#960;«/mo»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math», in order to find a function modelling the area of the spill as a function of time. To do this, it is necessary to introduce some new notation. 

The result of this type of combination is called a composite function.

In this notation, the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is substituted into the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mstyle»«/math» in place of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math».

Applying this notation to the Deepwater Horizon oil spill model, you can see the radius function replaces «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»r«/mi»«/mstyle»«/math» in «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»A«/mi»«mo»=«/mo»«mo»§#960;«/mo»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math».

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