U1A L4 Inverse Functions Skill Builder
Completion requirements
Unit 1A
Precalculus
Lesson 4: Transformations
Skill Builder
Domain and Range
For a function and its inverse, the domain of the function becomes the range of the inverse. Likewise, the range of the function becomes the domain of the inverse.a.
Graph 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»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math».
b.
Use key points to map the graph of the inverse on the same grid.
c.
Show the reflection line and state any invariant points.
d.
State the domain and range of both relations.
a.
b.
Use the key points «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»4«/mn»«/mrow»«/mfenced»«mi
mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»0«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math», and «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
To determine the points on the graph of the inverse of the function, switch the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»- and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values for each point.
The graph of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» and its inverse «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msup»«mi»f«/mi»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/msup»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» are shown below.
To determine the points on the graph of the inverse of the function, switch the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»- and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values for each point.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨left¨»«mtr»«mtd»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»4«/mn»«/mrow»«/mfenced»«mo»§#8594;«/mo»«mfenced»«mrow»«mn»4«/mn»«mi
mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mo»§#8722;«/mo»«mn»4«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»0«/mn»«/mrow»«/mfenced»«mo»§#8594;«/mo»«mfenced»«mrow»«mn»0«/mn»«mi
mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mfenced»«mrow»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»4«/mn»«/mrow»«/mfenced»«mo»§#8594;«/mo»«mfenced»«mrow»«mn»4«/mn»«mi
mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»0«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The graph of the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» and its inverse «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msup»«mi»f«/mi»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/msup»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» are shown below.
c.
The equation of the line of reflection is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math». It is shown by the black
dashed line.
The invariant points would occur on the reflection line. There are no invariant points.
The invariant points would occur on the reflection line. There are no invariant points.
d.
The domain of 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»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/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»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
The range of 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»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/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»y«/mi»«mo»§#8739;«/mo»«mi»y«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»y«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
The domain of the inverse 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»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
The range of the inverse is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»y«/mi»«mo»§#8739;«/mo»«mi»y«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
The inverse relation is not a function.
The range of 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»«msup»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/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»y«/mi»«mo»§#8739;«/mo»«mi»y«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»y«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
The domain of the inverse 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»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
The range of the inverse is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mi»y«/mi»«mo»§#8739;«/mo»«mi»y«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»R«/mi»«/mrow»«/mfenced»«/mstyle»«/math».
The inverse relation is not a function.
Determining the Equation of the Inverse
Mapping is a useful technique to determine the graph of the inverse of a function. However, it is not the only way to graph the inverse of a function. If the equation of the inverse can be determined, technology can be used to graph the inverse of the function.When you were given a list of points on the graph of a function, or when you were given a graph from which to read points, you ‘switched’ the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»- and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-coordinates to determine the corresponding points on the graph of the function’s inverse.
Given the equation of the function, the equation of the inverse can be determined by switching the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» variables in the original equation, and then solving for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math».
a.
Algebraically determine the equation of the inverse of 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»«mfrac»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»8«/mn»«/mrow»«mn»3«/mn»«/mfrac»«/mrow»«/mstyle»«/math».
b.
Verify the equation graphically.
a.
Rewrite «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»«mfrac»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»8«/mn»«/mrow»«mn»3«/mn»«/mfrac»«/mrow»«/mstyle»«/math»
as an equation in the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math».
Now, “switch” the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» variables in the equation.
Then, solve for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math».
The equation of the inverse of the function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»8«/mn»«/mrow»«/mstyle»«/math» or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»f«/mi»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/msup»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»8«/mn»«/mrow»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»8«/mn»«/mrow»«mn»3«/mn»«/mfrac»«/mrow»«/mstyle»«/math»
Now, “switch” the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» variables in the equation.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»y«/mi»«mo»+«/mo»«mn»8«/mn»«/mrow»«mn»3«/mn»«/mfrac»«/mrow»«/mstyle»«/math»
Then, solve for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/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»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»y«/mi»«mo»+«/mo»«mn»8«/mn»«/mrow»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mn»3«/mn»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»y«/mi»«mo»+«/mo»«mn»8«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»3«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»8«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»y«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The equation of the inverse of the function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»8«/mn»«/mrow»«/mstyle»«/math» or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»f«/mi»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/msup»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»8«/mn»«/mrow»«/mstyle»«/math».
b.
The graph of both the function and its inverse should be reflections in the line «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math».
Any point on the graph of the inverse should also correspond to a point on the graph of the original function with its «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-
and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-coordinates switched.
Any invariant points will be on the line «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math».
The point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math» lies on both the graph of the original function and on the graph of its inverse. The point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math» also lies on the line of reflection, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math». Point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math» is an invariant point.
Point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»3«/mn»«/mrow»«/mfenced»«/mstyle»«/math» is on the graph of the function, and the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math» is on the graph of the inverse.
Therefore, the graph verifies the equation of the inverse.
Any invariant points will be on the line «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math».
The point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math» lies on both the graph of the original function and on the graph of its inverse. The point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math» also lies on the line of reflection, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math». Point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math» is an invariant point.
Point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»3«/mn»«/mrow»«/mfenced»«/mstyle»«/math» is on the graph of the function, and the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math» is on the graph of the inverse.
Therefore, the graph verifies the equation of the inverse.