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InverseFunctions
One-to-one
Supposef:A→Bisafunct ion .Wecallfone-to-oneifeveryd istinct pairofobjectsi nAisassign edtoadistinctpairo fob jectsi nB.Ino ther words,eachobjectof thetargetha satmostoneobjectfr omthed omain assignedto it. Therearetwoeq uivalent waysofphr asingthepreviousdefinitionin more mathematicallanguage.Thefirstistosay thatfisone-to -oneifwhenever weha vetwoobjectsa,c?Awitha?=c,we aregua ranteedthat f(a)?=f(c). Thesecon d,istosaythatfbeingone-to-onem eansthatifweeverhave objectsw,z?Awithf(w)=f(z),then itmustb ethatw=z. Example.g:R→Rwhereg(x)=x+3i son e-to-o ne.Tocheckthis, wesupp osethatw,z?Raresuch thatg(w)=g(z).The nw+3=z+3.
Subtract3frombothsidesoft heeq uation,a ndw=z.
Weshowed thatifg(w)=g(z),then w=z.Tha tmeansthat gisone-to - one.
Horizontallinetest
Ifah orizon tallineintersectsthegraph off(x)inmorethanonepoint, thenf(x)isnotone-to-one. twopoints thathavethesamese condcoordinat e-forexampl e,(2,3)and (4,3).Thatw ouldmeanthat f(2)andf(4)b othequal3,andone -to-one functionscan'tassigntwodifferentobjectsint hedomaintothesameobje ct ofthetarget.
IfeveryhorizontallineinR
2 intersectsthegraphofafunctionatmost once,t henthefuncti onisone-to-on e. Onto Supposef:A→Bisafunct ion .Wecallfontoiftherange offequals B. Inoth erwords,fisonto ifeveryobjectint hetarg ethasatlea stoneobject fromthedomain assignedt oitbyf. 1 Example.h:R!Rwhereh(x) = 2xis onto. To check this, we need to choose a random object from the target ofh, and show that there is an object in the domain thathassigns to it. Here's how we do that:
Lety2R. (Notice thatRis the target ofh.) Theny2
is an object in the domain ofh, sinceRis the domain ofhandy2
2R. Nowh(y2
) = 2y2 =y, so y2 is an object in the domain ofhthat is assigned toybyh.
What an inverse function is
Supposef:A!Bis a function. A functiong:B!Ais called the inverse functionoffiffg=idandgf=id. Ifgis the inverse function off, then we often renamegasf1.
Examples.
Letf:R!Rbe the function dened byf(x) =x+ 3, and let g:R!Rbe the function dened byg(x) =x3. Then fg(x) =f(g(x)) =f(x3) = (x3) + 3 =x Becausefg(x) =xandid(x) =x, these are the same function. In symbols, fg=id.
Similarly
gf(x) =g(f(x)) =g(x+ 3) = (x+ 3)3 =x sogf=id. Therefore,gis the inverse function off, so we can renameg asf1, which means thatf1(x) =x3. Letf:R!Rbe the function dened byf(x) = 2x+ 2, and let g:R!Rbe the function dened byg(x) =12 x1. Then fg(x) =f(g(x)) =f12 x1 = 212 x1 + 2 =x
Similarly
gf(x) =g(f(x)) =g(2x+ 2) =12 2x+ 2 1 =x Therefore,gis the inverse function off, which means thatf1(x) =12 x1. 2
The Inverse of an inverse is the original
Iff1is the inverse off, thenf1f=idandff1=id. We can see from the denition of inverse functions above, thatfis the inverse off1.
That is (f1)1=f.
Inverse functions \reverse the assignment"
The denition of an inverse function is given above, but the essence of an inverse function is that it reverses the assignment dictated by the original function. Iffassignsatob, thenf1will assignbtoa. Here's why: Iff(a) =b, then we can applyf1to both sides of the equation to obtain the new equationf1(f(a)) =f1(b). The left side of the previous equation involves function composition,f1(f(a)) =f1f(a), andf1f=id, so we are left withf1(b) =id(a) =a. The above paragraph can be summarized as \Iff(a) =b, thenf1(b) =a."
Examples.
Iff(3) = 4, thenf1(4) = 3.
Iff(2) = 16, thenf1(16) =2.
Iff(x+ 7) =1, thenf1(1) =x+ 7.
Iff1(0) =4, thenf(4) = 0.
Iff1(x23x+ 5) = 3, thenf(3) =x23x+ 5.
When a function has an inverse
A function has an inverse exactly when it isbothone-to-one and onto. This will be explained in more detail during lecture.
Examples.
It was shown earlier thatg:R!Rwhereg(x) =x+3 is one-to-one. You can also check thatgis onto. Therefore,ghas an inverse function,g1. It was shown earlier thath:R!Rwhereh(x) = 2xis onto. You can check thathis one-to-one as well, which means that there is an inverse function,h1. 3
Using inverse functions
Inverse functions are useful in that they allow you to \undo" a function. Below are some rather abstract (though important) examples. As the semes- ter continues, we'll see some more concrete examples.
Examples.
Suppose there is an object in the domain of a functionf, and that this object is nameda. Suppose that you knowf(a) = 15. Iffhas an inverse function,f1, and you happen to know thatf1(15) = 3, then you can solve foraas follows:f(a) = 15 implies thatf1(15) =a. Thus, a= 3. Ifbis an object of the domain ofg,ghas an inverse,g(b) = 6, and g
1(6) =2, then
2 =g1(6) =b
Supposef(x+ 3) = 2. Iffhas an inverse, andf1(2) = 7, then x+ 3 =f1(2) = 7 so x= 73 = 4
The Graph of an inverse
Iffis aninvertiblefunction (that means iffhas an inverse function), and if you know what the graph offlooks like, then you can draw the graph of f 1. If (a;b) is a point in the graph off(x), thenf(a) =b. Hence,f1(b) =a. That meansf1assignsbtoa, so (b;a) is a point in the graph off1(x). Geometrically, if you switch all the rst and second coordinates of points inR2, the result is to ipR2over the \x=yline". 4
NewHowpoin tsingraphoff(x)visualeffect
functionbecomepointsof newgraph f -1 (x)(a,b)?→(b,a)flipoverth e"x=yline"
Examples.
5
How to nd an inverse
If you know thatfis an invertible function, and you have an equation for f(x), then you can nd the equation forf1in three steps.
Step 1is to replacef(x) with the lettery.
Step 2is to replace everyxwithf1(y).
Step 3is to use algebra to solve forf1(y).
After using these three steps, you'll have an equation for the function f 1(y).
Examples.
Find the inverse off(x) =x+ 5.
Step 1.y=x+ 5
Step 2.y=f1(y) + 5
Step 3.f1(y) =y5
Find the inverse ofg(x) =2xx1.
Step 1.y=2xx1
Step 2.y=2g1(y)g
1(y)1
Step 3.g1(y) =yy2
Make sure that you are comfortable with the algebra required to carry out step 3 in the above problem. You will be expected to perform similar algebra on future exams. You should also be able to check thatgg1=idand thatg1g=id. 6
Exercises
In #1-6,gis an invertible function.
1.) Ifg(2) = 3, what isg1(3)?
2.) Ifg(7) =2, what isg1(2)?
3.) Ifg(10) = 5, what isg1(5)?
4.) Ifg1(6) = 8, what isg(8)?
5.) Ifg1(0) = 9, what isg(9)?
6.) Ifg1(4) = 13, what isg(13)?
For #7-12, solve forx. Use thatfis an invertible function and that f
1(1) =2
f
1(2) = 3
f
1(3) = 2
f
1(4) = 5
f
1(5) =7
f
1(6) = 8
f
1(7) =3
f
1(8) = 1
f
1(9) = 4
7.)f(x+ 2) = 5
8.)f(3x4) = 3
9.)f(5x) = 1
10.)f(2x) = 2
11.)f(1x
) = 8
12.)f(5x1) = 3
7 Each of the functions given in #13-18 is invertible. Find the equations for their inverse functions.
13.)f(x) = 3x+ 2
14.)g(x) =x+ 5
15.)h(x) =1x
16.)f(x) =xx1
17.)g(x) =2x+3x
18.)h(x) =x4x8
19.) Below are the graphs off(x) andf1(x). What are the coordinates
of the points A and B on the graph off1(x).
20.) Below are the graphs ofg(x) andg1(x). What are the coordinates of
the points C and D on the graph ofg(x).
9Exercises
In#1-6, gisanin vertib lefunction.
1.)If g(2)=3, what isg
-1 (3)?
2.)Ifg(7)= -2,whatis g
-1 (-2)?
3.)Ifg(-10)= 5,whatis g
-1 (5)?
4.)Ifg
-1 (6)= 8,what isg(8)?
5.)Ifg
-1 (0)= 9,what isg(9)?
6.)Ifg
-1 (4)= 13,what isg(13)? For#7-12,so lveforx.Use thatfisanin verti blefunctionandthat f -1 (1)= -2 f -1 (2)= 3 f -1 (3)= 2 f -1 (4)= 5 f -1 (5)= -7quotesdbs_dbs11.pdfusesText_17
FONCTION INVERSE - maths et tiques