One - to - One FunctionsNumerical, Graphical, and Analytical Approaches

A function is a. �0\� �� �

?( -\J� ��r� w·,� Ow'\1

� � Suck M' eo..�

!... t\--J�. If the point (2, -3) is on the graph of a function,.f{x), then no other point can have a(n) ?(-,J� of�.

However, it is possible for another point on the graph off(x) to have ay- coordinate of-3. For example, both (2, -3) and (5, -3).

A one-to-one function is 0... �CW"\ . \V\.. u.)�� .utt.h.. '3- .J� �c;

� � �-'1�.For example, if (2, -3) and (5, -3) are both on the graph off(x), then there is ay value, namely-3, that has more than one x value. Therefore, there is no way that the function is a one - to - one function.

(GraphkalJ)eterminationif a function is a one-to-

one function.

The inverse of a function is defined to be � �c..:i\.;� � . � �

� � � � A. �"" cur, sw·,.w...J ·What property of the function must exist in order for the inverse of the function to exist? Why? :!.� +()o.) \� � \-\ I ����4 '1-'1� 1 -r�) wtr\.,..l.l 'ct_� �.u!<.cl �-v� { .p-·c--�. �., �',&.) \"V\v<;.t be... \-\. �f �-,c)(.) \s +o

eA�-t.

If it can be detennined that a function is one-to-one, then the inverse of the function can be found. Fill in the following chart which will describe how to find the inverse of the function if it is established to be a one-to-one function.

function

Graphical Determination

of the inverse of a

function

Analytical Determination

of the inverse of a

function

Examples: For each of the following functions, determine if the inverse of the function exists, providing justification. If it does, then find the inverse. If the relation is given in numerical fonn, give the table of values that would representf-'(x). If the relation is given in graphical fom1, sketch the graph off-'(x). If the relation is given in analytical fonn, find the equation off-'(x).

a.

c.+'{>') ,� \-\ b\c:.. Graph of.f(x)

-It,._ i't-.�qa E I l I I l\t ·tl�\c_ �l!Nt l!:t!:I+� �-, � + ... :f. .... + ... :?. .... 1-1.· tt,.ffi .. 1 ... .A ...... + . . A-s s) (, -s" 1 ........ 1 ........ i .... ....1 .... . !-2- ..... 1 ...... 1 ..... . i ........ 1 ........ 1

J ) >) [ ....... ,i. ...... J ....... : .... ... i-3 ..... .i. .... .. ' .... � . .... .!. ....... ;2 ,) ( ;-"\, ' ' ' ' ' ' ' ' '

- ) \, -2 .1 t· .... --1·······-t-····--:--·· -t-4 ······t······· !---'····t · : ....... l . � - f) (-\) �) , ... .... i ..... L .. . . ; .... ...• _5 ............... i ........ •:,f��)

b.

I F;x)1-: I �2 1 �l I : I : I 'l=(>c.) , � \ -\ b\c � cMA. "'"'°

�� � -'1�. F-'c,..)e.-A�� b\c.. F�) , (, \ - \ .

' ' -2. -\ 2..

\ 5

d. )C.)i.� l-\ Graph of g(x)

l

� ! c .. ) i .. r-t 111-, i j N , V\0-1- I -I .

1: i j I �i:: ;,,i ::::::::,.

11':

.

;:',,, .,:�;.: t······r·····t·······r···r"'4 ... \

........ : ........ . ....... : ........ · 5- ······•···· .............. ,

""�), t:, \-\ b\c � e.� �� ....... 1 ...... m

.5���.�p\o� .. �.��! .... , ........ , ....... , f. j(x) = (x- 3)2

- 2' L-'/.'\ ' ' ' ' ' ' ' ' -,t(}&.�' � \ \ \,c � -� S \-l t.. T. n \.�1·+····+·· ··Y···+·-4· ······t·······l········t·····---l········l '� - , . ,-.. � - '

-w�tt lolc. 11:i It' ti!ii �� .... U)� WhA'\i�f"'� lx-) i c, , -\ · r·······rtr· ··T· ····r-1 ······r-···-r·····r·· ··r·····1 � \--\'-T • .P -- '(>c., � � "'� �'� r--·· ·-;-···r···-r···--r+ · ···-r-······i·······t···· .. ·1········1 �"-+ �le. � (� ·, � � \ - , . � 3). (: � -5) -�/-f ..... f. ..... -f .... i1. . ..... L .... t .. A .. . . +. ..... •

-\, s (.�,-i)

. y.. -:: 3) - 'i . 2.

�)( ':. ?,y - .,

�')(.'t � : 1'1

y=

g-'

6,.) :.

Find g- 1 (g(x)).

j<.f'"')) = ��� 4 f'�,��.a(�>4}+-=i 2.

3

: 2.��4-"\ -

'3 )( -'-\ � '{-

2. 3

;� ...

2 ')C.

�

..... � -

-

.,.. -

If two functions,.f{x) and g(x), are inverses of each other, then there are two special composite functions that will equal the same thing: -=-

Examples: Determine if each of the pairs of functions below are inverse functions of each other or not. Show your wor:�-�......,

-

..

X •

b . .f{x) = - -1 2

g(x) = E)

-t(��� -:. �)(.t\ - \::z.

-= ')(-+��--, - " J.. - "-::2.

s,�u.. �(s<>-$) =f= X, �

�(,--) �(},.) OW2. � l;t\\JUt,(.,? i .eA� �.

Name ��w� K� Period �������� ��-Date

Day #13 Homework

1. Complete the table below so that G(x) is a function, but is NOT a one-to-one function. Specificallyexplain why your example is not a one-to-one

rn:c� -\.....1,,k S,�� 'oe. u-f lU�

I G;x) 1-Z I O

1-l I O 14

I �!t:t � '4 �+ -e

2. Given that fix) and g(x) are two one-to-one functions and that fi..2) = 3 and g(3) = 1. Are f and ginverses of each other or not? Explain why or why not?

1.. l2. , . � -.:c:f -f(2..):; 3 o-wA.. �(&):' J � � f)Ol� I�...., \':. � • �� � �v-) � ( 3,,) i'> � � �� � �c.,c.). 6v+J h l � ,f � � w,e..v-e. \"\\J �� � � X : 3 =.,l � t"llf1� A. , � "4 _..., o..lM.t.. , 6 \o e ;2. . "'fl,w� ., f. � �o T

3. Anijlyti'edlly determme tffi..x) = 2x + 1 and g(x) = 0.5x - 1 are inverses of each other or not. Show '"'"e,s.ra.,your work and explain your reasoning based on that work.

� � ��c. .... )) = '2..(o. sx.-1) + \ �'"'� �(��)) 4, x ) � � o�.

: "X -:l +\ � � � � ��"�· � � +o Pc. '"""Q.\'"s..t4,

= � - \ � 4(.�c.�:)) � &(TO&-)) MU�"- � J. ?t. •

4. Graphed below are two functions, F and G. Determine if p-l andTor G-1 exist or not. Explain whywhy or why not. If the inverse does exist, then also draw a graph of the inverse on the grid.

F(x)

....... ·······.--······,········.--······,···5 ······,········.--······,········.--······,········, !·······+ .... ! ·······t'"······!········t···5· ·····+·······l·······+······-1-···· .. ·l······"! 1. .... ...1 ...... l ..... l ....... I ....... 1 .. :. . ..... l... .... J ........ 1... .. ...1. .... ...1 ....... J1 ...... J ....... , ...... + ....... j ....... : - · : ...... 1 ...... J ....... 1 ........ 1 ....... 1 i·· .. ····t· .. ····Y·····t·· .... ··i··· .. ···t·+ ·····'!' ··· : ·······t····· .. ·i········r· .... i

� � � � � � j � j 4 j • r·······r········r·······r········i········r-l· ······r······· · ····r· .... · .. i········r ........ ,

�+-i-+-!-1� ll. l-.i-t}) l l � l 1 1 1 1 i : : l !········!········�······ .. +········f ........ +-5· ······+········!········!········!········+····· i. ....... i ..... ...i. ....... : ...... ..i.. ...... i_5 ...... i ....... .i... ..... : ..... ...i. ....... : .... ....i

l=--, l y.') � �" t b\c. �C,C..) \ � Q.,.

\-\ �� �\�e. ·l{. ��

� V\.D �.3tJ'A..\..1. wu..�e.s.+. f,�) (�,lo) {-la oa) ('J., � (� -t.t) (r. -s

r'C1-) (� CJ.-, t, <.� 3 e:-sJ�

For problems 5 - 8, use a graphing calculator to draw a rough sketch of the graph of the function given. Draw a sketch of the graph on a set of axes. Then, (a) determine if the function is a one-to-one functionor not, explaining your reasoning for each one and (b) if the function is one-to-one, find the equation ofthe inverse of the function. Show all of your work and use complete sentences in your statements ofreasoning.

5. f(x) = 3x + 1

2

p x:: �2.

:l.� :: �� � \

�� -\ ':= 3� � ':: l� -\

3

7. f(x) =x

-3

x+2

� ':'. .�-3 8""�

')(. � -\- 2.. ')( .:: � -3 x�-J=-�-3

J('X.-1): -2.x-3 j -:. -2.x.-3 or- _ 2.JC.-+3

)(.-\ �-,

.;-•c,.) = - ;l<.�f o..-�i"O.

�\(J'r\

-,"L.t. � r+ 1, �) d � """-*

y�c.s � V\&f';o�"-.0 W\L -+e.,+ • �S I �(.,-_) \ \ J

\-\ �c.l,\, ��

-+ - ' (.--.) � � 0t1s.-'c .

@) ){: -J� -\-'S. -t4\­.,. - � : :- r� � -i.-.

� -�) L. :: 'd -to :l..

�: �- Ui):l,.

-r � ) -x � "1

' -r--'oc-) ': (� -'-l)l.

-1. ) � ��

f-t�J � �'1.-8)( -t\4, � �1