The Way of the Inverse!

By Sean PrinsFor: Education 205

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Inverse’sWhat is an inverse?

Examples of how to get an inverse.Graphing inverse!Video to help you graph inverse!

Definition for two inverses being inverses of each other. Examples of two inverses being inverses of each oth

er.

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What is an inverseWhen you are trying to find an inverse for an

equation, you have to switch the x’s with the y’s and the y’s with the x’s.

-Example one-Example two

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Example 1So you have the equation.

y=2x-5First you switch the x and the y.

x=2y-5Then you solve for y.

x=2y-5 First add 5 on both sides. x+5=2y-5+5 The 5’s on the right side cancel

out. x+5=2y Multiply by ½ on both sides.½(x+5)=(2y)1/2 The 2 and the ½ cancel out on

the right side(x+5)/2=y And that is the inverse!

Click here for example 2End Show

Example 2So you have the equation.

y=x ² +5First you switch the x and the y.

x=y ² +5Then you solve for y.

x=y ² +5 Subtract 5 from both sides x-5=y ² +5-5 The 5 by the y cancel out x-5=y ² Then square root both sides√x-5 = √y² The squared and the square root

cancel out.√x-5=y There you have it, that’s the

answer!End Show

Graph of the inverse.I am going to use the example that I used in

example one. The original equation is in red and the

inverse is in black.Do you see a connection

between the two functions?Click here to find out the

answer!

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AnswerYes these is a connection. The inverse is flipped

over the y=x axis.

But just because the inverse can be graphed, it does not mean it is a function. You still have to do the Horizontal Line Test. So the graph on the left would fall the HLT.End

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Video of Graphing!Click here for the video of solving and graphi

ng an inverse function. This should be really good!

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How do you check to see if two graphs are inverses of

each other?You will be given to equations. Lets say you

are given f(x) and g(x). The only way you can tell they are inverses is if you take one and substitute it for the other. For instance, you need to take g(x) and put it into f(x). This is what it looks like. f(g(x)). Then you have to do the opposite. g(f(x)). The only way they are inverses is if f(g(x)) and g(f(x)) equals x.

Example 1Example 2

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Example 1Lets use equations that we already know are inverses. f(x)=2x-5 and g(x)=(x+5)*1/2. f(g(x))=2((x+5)*1/2)-5 The 2 and ½ cancel out.f(g(x))=(x+5)-5 The 5 and -5 cancel out.f(g(x))=xNow for g(f(x))g(f(x))=((2x-5)+5)* ½ The 5 cancel out leavingg(f(x))=(2x)*1/2 The 2 and ½ cancel outg(f(x))= xg(f(x))=f(g(x)) They are inverses.

Click here for another example! End Show

Example 2Now for my second example, I am going to

give you the problems and you are going to have to try to solve it. The answer will be in the link below, only if you have any trouble with it.

f(x)=x³+8g(x)=³√(x)-2

Answer here!End Show

AnswerYes they are inverses of each other. f(x)=x³+8 and g(x)=³√(x)+2f(g(x))= (³√(x)-2)³+8 You have to distibute

the third power to the ³√(x) and the 2.

f(g(x))= ³√(x) ³-(2) ³+8 The third power and the ³√ cancel out.

f(g(x))=x-8+8f(g(x))=xClick here for g(f(x))!End

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Answer 2 for example 2!f(x)= x³+8 and g(x)=³√(x)-2. This is what you

need to do for g(f(x)).g(f(x))=³√(x³+8)-2 This is what you need to

start out with.g(f(x))=³√(x³)+ ³√(8)-2 You have to distibute

the cube root to x³ and the 8.g(f(x))=x+2-2 Cancel out the cube rood

and the third power.g(f(x))=x g(f(x)) equals the same as

f(g(x)). They are inverses!End Show

Authors Page

Sean Prins is a student at Grand ValleyState University. He is currently trying to get his major in Mathematics and has not chosen a minor yet. He tries to be very active. He plays hockey almost every week and enjoys fishing on the weekends. If he is not in school, like

during the summer and winter break, he is working many hours a week at his job.

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Resource slidehttp://www.paly.net/~sfriedla/algebratwo/Not

es/Unit7/AlgIINotes7_4.html

http://regentsprep.org/Regents/mathb/3d4/applesson.htm

http://faculty.eicc.edu/bwood/ma155supplemental/supplemental3.htm

www.teachertube.comEnd Show

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