L5.1 C2 Composite Functions - Part 2
Completion requirements
Unit 5
Function Operations
Order of Composition
The order of composition matters.
If a function is substituted into a second function, the result is not the same as when the second function is substituted into the first.
In Example 1, you determined the equation of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»f«/mi»«mo»§#8728;«/mo»«mi»g«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»13«/mn»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math» and «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»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math». In the next Example, you will determine the equation of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»g«/mi»«mo»§#8728;«/mo»«mi»f«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math».
In Example 1, you determined the equation of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»f«/mi»«mo»§#8728;«/mo»«mi»g«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»6«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»13«/mn»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math» and «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»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math». In the next Example, you will determine the equation of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»g«/mi»«mo»§#8728;«/mo»«mi»f«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math».
Question
Given «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math» and «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»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math», determine «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»g«/mi»«mo»§#8728;«/mo»«mi»f«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math».
The notation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»g«/mi»«mo»§#8728;«/mo»«mi»f«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» indicates the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» will be substituted into 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».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»g«/mi»«mo»§#8728;«/mo»«mi»f«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»g«/mi»«mfenced»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math»The «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» in «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» will be replaced by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»g«/mi»«mfenced»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mfenced»«mrow»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»§#8722;«/mo»«mn»5«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«mo»§#8722;«/mo»«mn»5«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»6«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»The algebraic result of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»g«/mi»«mo»§#8728;«/mo»«mi»f«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» is not the same as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»f«/mi»«mo»§#8728;«/mo»«mi»g«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math».
Variable Restrictions
If a function has restrictions on its domain, and that function is substituted into another, the restrictions carry into the composite function.
Refer to Example 2 on page 502 of Pre-Calculus 12.
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»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math» has no restrictions on its domain.
The function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msqrt»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/msqrt»«/mrow»«/mstyle»«/math» has a restriction. The value under the root sign must be greater than or equal to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math».
So, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8805;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» is the restriction.
The possible composite functions are as follows.
At first glance, there don't appear to be any variable restrictions for the composite function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»g«/mi»«mo»§#8728;«/mo»«mi»f«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math». However, the fact a radical was embedded in the composition means the variable restrictions for that radical carry through and apply to the composite function.
The idea of attending to the variable restrictions on the individual functions before the composite function is determined is similar to the idea of stating the variable restrictions of a rational function before it is simplified.
As such, the domain of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»g«/mi»«mo»§#8728;«/mo»«mi»f«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mfenced»«msqrt»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«mn»2«/mn»«/msup»«mo»=«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»x«/mi»«mo»§#8739;«/mo»«mi»x«/mi»«mo»§#8805;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
Refer to Example 2 on page 502 of Pre-Calculus 12.
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»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math» has no restrictions on its domain.
The function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msqrt»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/msqrt»«/mrow»«/mstyle»«/math» has a restriction. The value under the root sign must be greater than or equal to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math».
So, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8805;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» is the restriction.
The possible composite functions are as follows.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»f«/mi»«mo»§#8728;«/mo»«mi»g«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»1«/mn»«/msqrt»«/mrow»«/mstyle»«/math»And
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfenced»«mrow»«mi»g«/mi»«mo»§#8728;«/mo»«mi»f«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«msqrt»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»The variable restrictions for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»f«/mi»«mo»§#8728;«/mo»«mi»g«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» are as follows.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»1«/mn»«/mtd»«mtd»«mo»§#8805;«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»§#8805;«/mo»«/mtd»«mtd»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mfenced open=¨|¨ close=¨|¨»«mi»x«/mi»«/mfenced»«/mtd»«mtd»«mo»§#8805;«/mo»«/mtd»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8804;«/mo»«mo»§#8722;«/mo»«mn»1«/mn»«mspace width=¨0.33em¨/»«mi»or«/mi»«mspace width=¨0.33em¨/»«mi»x«/mi»«mo»§#8805;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math»The restrictions on this composite function are more restrictive than those on the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math», so they are kept.
At first glance, there don't appear to be any variable restrictions for the composite function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»g«/mi»«mo»§#8728;«/mo»«mi»f«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math». However, the fact a radical was embedded in the composition means the variable restrictions for that radical carry through and apply to the composite function.
The idea of attending to the variable restrictions on the individual functions before the composite function is determined is similar to the idea of stating the variable restrictions of a rational function before it is simplified.
As such, the domain of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»g«/mi»«mo»§#8728;«/mo»«mi»f«/mi»«/mrow»«/mfenced»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mfenced»«msqrt»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/msqrt»«/mfenced»«mn»2«/mn»«/msup»«mo»=«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»x«/mi»«mo»§#8739;«/mo»«mi»x«/mi»«mo»§#8805;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».