GRE Math Question (Set 1 of 2 Explained)

Page 1

í »

7KH DPRXQW RI 7KH QXPHUDWRU GLYLGHG E\ UDWLRQDO QXPEHUV LQ WKH GHQRPLQDWRU WKH IUDFWLRQ DERYH

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU $

([SODQDWLRQ

$ IUDFWLRQ LV D QXPEHU RI WKH IRUP D»E ZKHUH D DQG E DUH LQWHJHUV DQG E 7KH LQWHJHU D LV FDOOHG WKH QXPHUDWRU RI WKH IUDFWLRQ DQG E LV FDOOHG WKH GHQRPLQDWRU )RU H[DPSOH í LV D IUDFWLRQ LQ ZKLFK LV WKH QXPHUDWRU DQG LV WKH GHQRPLQDWRU 6XFK QXPEHUV DUH DOVR FDOOHG UDWLRQDO QXPEHUV 7KHUH DUH UDWLRQDO QXPEHUV LQ WKH IUDFWLRQ í í LV D UDWLRQDO QXPEHU DQG LV D UDWLRQDO QXPEHU 7KH QXPHUDWRU GLYLGHG E\ WKH GHQRPLQDWRU LV

([DPSOH RI KRZ WKH OHIW VLGH PD\ EH ZURQJO\ LQWHUSUHWHG LV D UDWLRQDO QXPEHU EHFDXVH LW FDQ EH ZULWWHQ DV D IUDFWLRQ

+HUH DUH VRPH PRUH H[DPSOHV 1XPEHU

$V D )UDFWLRQ

5DWLRQDO"

<HV

<HV

<HV

<HV

<HV

Â¥ VTXDUH URRW RI

"

12

2RSV 7KH VTXDUH URRW RI FDQQRW EH ZULWWHQ DV D VLPSOH IUDFWLRQ $QG WKHUH DUH PDQ\ PRUH VXFK QXPEHUV DQG EHFDXVH WKH\ DUH QRW UDWLRQDO WKH\ DUH FDOOHG ,UUDWLRQDO $QRWKHU IDPRXV LUUDWLRQDO QXPEHU LV 3L ʌ

í »

í » »

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU %

([SODQDWLRQ

'LVUHJDUG WKH QHJDWLYH WHPSRUDULO\ \RX¶UH PXOWLSO\LQJ RQH YDOXH E\ DQG WKH RWKHU YDOXH E\ 0XOWLSO\LQJ E\ LQFUHDVHV WKH YDOXH PRUH WKDQ PXOWLSO\LQJ E\ 7KXV LI WKH TXHVWLRQ ZDV QRW XVLQJ QHJDWLYHV ZH ZRXOG FKRRVH $ ,QWURGXFH WKH QHJDWLYH LQWR \RXU WKLQNLQJ E\ VZLWFKLQJ \RXU DQVZHU IURP $ WR %

í í í U

í í í U

U U

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU &

([SODQDWLRQ

,I ERWK WKH QXPHUDWRU DQG GHQRPLQDWRU KDYH D FRPPRQ IDFWRU WKHQ WKH QXPHUDWRU DQG GHQRPLQDWRU FDQ EH IDFWRUHG DQG UHGXFHG WR DQ HTXLYDOHQW IUDFWLRQ )RU H[DPSOH » 7KLV LV DQ H[DPSOH RI PDNLQJ WKH GHQRPLQDWRUV HTXDO ZKHQ FRPSDULQJ IUDFWLRQV

í Ô Ô

Ô í Ô

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU %

([SODQDWLRQ 7R DGG WZR IUDFWLRQV ZLWK WKH VDPH GHQRPLQDWRU \RX DGG WKH QXPHUDWRUV DQG NHHS WKH VDPH GHQRPLQDWRU )RU H[DPSOH » í

7R DGG WZR IUDFWLRQV ZLWK GLIIHUHQW GHQRPLQDWRUV ILUVW ILQG D FRPPRQ GHQRPLQDWRU ZKLFK LV D FRPPRQ PXOWLSOH RI WKH WZR GHQRPLQDWRUV 7KHQ FRQYHUW ERWK IUDFWLRQV WR HTXLYDOHQW IUDFWLRQV ZLWK WKH VDPH GHQRPLQDWRU )LQDOO\ DGG WKH QXPHUDWRUV DQG NHHS WKH FRPPRQ GHQRPLQDWRU

)RU H[DPSOH WR DGG WKH IUDFWLRQV DQG í XVH WKH FRPPRQ GHQRPLQDWRU í í í

í í í í

:KLFK RI WKH IROORZLQJ LV D )$/6( VWDWHPHQW"

$ $Q H[SUHVVLRQ VXFK DV ñ ೪ LV FDOOHG D PL[HG QXPEHU % ñ ೪ FRQVLVWV RI DQ LQWHJHU SDUW DQG D IUDFWLRQ SDUW & ñ ೪ PHDQV ñ ೪ ' 7R FRQYHUW ñ ೪ WR DQ RUGLQDU\ IUDFWLRQ FRQYHUW WKH LQWHJHU SDUW WR DQ HTXLYDOHQW IUDFWLRQ DQG DGG LW WR WKH IUDFWLRQ SDUW ( 1XPEHUV RI WKH IRUP D E ZKHUH HLWKHU D RU E LV QRW DQ LQWHJHU DQG E DUH IUDFWLRQDO H[SUHVVLRQV WKDW FDQ EH PDQLSXODWHG MXVW OLNH IUDFWLRQV

&RUUHFW $QVZHU (

([SODQDWLRQ $QVZHU ( VKRXOG VD\ µ1XPEHUV RI WKH IRUP D E ^127 D E ` ZKHUH HLWKHU D RU E LV QRW DQ LQWHJHU DQG E DUH IUDFWLRQDO H[SUHVVLRQV WKDW FDQ EH PDQLSXODWHG MXVW OLNH IUDFWLRQV ¶

,I ZH ZDQWHG D VHQWHQFH XVLQJ D E WKHQ 1XPEHUV RI WKH IRUP D E ^127 D E ` DUH PXOWLSOLFDWLRQ H[SUHVVLRQV QRW IUDFWLRQDO H[SUHVVLRQV WKDW FDQ EH PDQLSXODWHG MXVW OLNH PXOWLSOLHG QXPEHUV

7KLV LV VLPSO\ H[SODLQLQJ WKH IRUP D E LV FDOOHG D IUDFWLRQ $QG E FDQQRW EH ]HUR ,I E LV ]HUR RXU IUDFWLRQ GRHV QRW KDYH D GHILQHG GHQRPLQDWRU 7KXV LI E ZDV ]HUR WKHQ RXU IUDFWLRQ LV XQGHILQHG DQG FDQQRW EH PDQLSXODWHG OLNH D IUDFWLRQ

OLNH D DQG E FDQ EH PDQLSXODWHG LQ WKRXJKW H[SHULPHQWV

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU &

([SODQDWLRQ

7R PXOWLSO\ WZR IUDFWLRQV ZH PXOWLSO\ QXPHUDWRUV RI ERWK IUDFWLRQV DQG WKHQ PXOWLSO\ GHQRPLQDWRUV RI ERWK IUDFWLRQV

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU $

&RUUHFW $QVZHU &

([SODQDWLRQ

7R PXOWLSO\ WZR IUDFWLRQV ZH PXOWLSO\ QXPHUDWRUV RI ERWK IUDFWLRQV DQG WKHQ PXOWLSO\ GHQRPLQDWRUV RI ERWK IUDFWLRQV

LV WKH UHFLSURFDO RI

!

¸

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU &

([SODQDWLRQ

7R GLYLGH WZR IUDFWLRQV ZH ILUVW IOLS RQH RI WKH IUDFWLRQV :LWK RQH IUDFWLRQ IOLSSHG DQG WKH RWKHU QRUPDO ZH WKHQ MXVW PXOWLSO\ QXPHUDWRUV RI ERWK IUDFWLRQV DQG WKHQ PXOWLSO\ GHQRPLQDWRUV RI ERWK IUDFWLRQV ¸

1RZ ZH KDYH WR FRPSDUH DQG :H ILQG D FRPPRQ GHQRPLQDWRU DQG ZH DUH FRPSDULQJ DQG

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU %

([SODQDWLRQ

7R GLYLGH WZR IUDFWLRQV ZH ILUVW IOLS RQH RI WKH IUDFWLRQV :LWK RQH IUDFWLRQ IOLSSHG DQG WKH RWKHU QRUPDO ZH WKHQ MXVW PXOWLSO\ QXPHUDWRUV RI ERWK IUDFWLRQV DQG WKHQ PXOWLSO\ GHQRPLQDWRUV RI ERWK IUDFWLRQV ¸

LV WKH UHFLSURFDO RI

!

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU $

([SODQDWLRQ

, ZLOO EH SUHFLVH , ZLOO VLPSOLI\ DFFXUDWHO\ 5HGXFH WKH IUDFWLRQV WKHQ PXOWLSO\ SUHFLVHO\ 7KHQ DGG IUDFWLRQV SUHFLVHO\ 7KHQ FRPSDUH SUHFLVHO\ t ೪

t ೪

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU &

, ZLOO EH SUHFLVH , ZLOO VLPSOLI\ DFFXUDWHO\ 5HGXFH WKH IUDFWLRQV WKHQ PXOWLSO\ SUHFLVHO\ 7KHQ DGG IUDFWLRQV SUHFLVHO\ 7KHQ FRPSDUH SUHFLVHO\ t ೪

t ೪

t ೪ t ೪

рин рин

рин

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU $

([SODQDWLRQ

7KLV LV ZKDW TXHVWLRQ ZDV H[SODLQLQJ WKDW рин FDQ EH WKRXJKW RI DV D QRUPDO IUDFWLRQ 7KXV WR DGG рин рин ZH FDQ VLPSO\ ILQG D FRPPRQ GHQRPLQDWRU DQG UHZULWH WKH QXPHUDWRUV ZLWK WKH FRHIILFLHQW RI FKDQJH )RU H[DPSOH рин рин рин рин рин рин рин

/HW┬╢V WU\ WKDW DJDLQ рин рин

рин ! рин

¥ ¥

¥ ¥

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU &

([SODQDWLRQ

,Q WKH VDPH ZD\ ZH WHPSRUDULO\ GLVUHJDUG QHJDWLYH VLJQV ZH FDQ WHPSRUDULO\ GLVUHJDUG WKH UDGLFDOV ZKLOH ZH JHW VLPSOLILHG DQG VHW XS ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ 'RQ¶W FRQYHUW WKH UDGLFDOV EHFDXVH \RX PD\ EH DEOH WR FRPSDUH DFFXUDWHO\ ZLWKRXW VROYLQJ ¥ ¥ ¥ ¥

рин ├▓рин

р╡│ рин

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU $

([SODQDWLRQ

7R DGG WZR IUDFWLRQV ZLWK GLIIHUHQW GHQRPLQDWRUV ILUVW ILQG D FRPPRQ GHQRPLQDWRU ZKLFK LV D FRPPRQ PXOWLSOH RI WKH WZR GHQRPLQDWRUV 7KHQ FRQYHUW ERWK IUDFWLRQV WR HTXLYDOHQW IUDFWLRQV ZLWK WKH VDPH GHQRPLQDWRU )LQDOO\ DGG WKH QXPHUDWRUV DQG NHHS WKH FRPPRQ GHQRPLQDWRU

)RU H[DPSOH WR DGG WKH IUDFWLRQV рин DQG ├▓рин XVH WKH FRPPRQ GHQRPLQDWRU рин ├▓рин рин ├▓рин рин рин рин рин рин

рин ! р╡│ рин

¥ ¥ ¥ ¥

¥ ¥

î ¥ ¥

$ 7KH TXDQWLW\ RQ WKH OHIW LV JUHDWHU % 7KH TXDQWLW\ RQ WKH ULJKW LV JUHDWHU & %RWK DUH HTXDO ' 7KH UHODWLRQVKLS FDQQRW EH GHWHUPLQHG ZLWKRXW IXUWKHU LQIRUPDWLRQ

&RUUHFW $QVZHU %

([SODQDWLRQ

,Q WKH VDPH ZD\ ZH WHPSRUDULO\ GLVUHJDUG QHJDWLYH VLJQV ZH FDQ WHPSRUDULO\ GLVUHJDUG WKH UDGLFDOV ZKLOH ZH JHW VLPSOLILHG DQG VHW XS ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ,Q WKLV FDVH FRQYHUW WKH UDGLFDOV EHFDXVH LQ WKLV FDVH WKH\ DUH HDV\

7R PXOWLSO\ IUDFWLRQV PXOWLSO\ WKH QXPHUDWRUV WRJHWKHU DQG WKHQ PXOWLSO\ WKH GHQRPLQDWRUV WRJHWKHU

¥ ¥

î ¥ ¥

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥

LV WKH UHFLSURFDO RI

!

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.