Ngā Kaupapa Taketake o te Mahi Whakamuri
I roto i te pāngarau, he ture te mahi e hono ana i ia huānga o tētahi huinga (te rohe) ki tētahi huānga o tētahi atu huinga (te rohe-ko). I roto i ngā ariā nui o ngā mahi, he tūranga taketake te mahi whakamuri nā te mea ka āwhina i a tātou ki te "whakahuri" i te tukanga hono. Mena ka hurihia e tētahi mahi he tāuru hei putanga, ko te whāinga o te mahi whakamuri—mēnā kei te wātea—ko te whakahoki i taua putanga ki te tāuru taketake. Ka matapakihia e tēnei tuhinga tōna whakamāramatanga, ngā tikanga mō te noho, me pēhea te tautuhi, tae atu ki ngā tauira me ngā tono.
1. Te Mārama ki ngā Mahi Whakamuri
Me kī he mahi \( f \) e hono ana i a \( x \) ki \( f(x) \). Ko te mahi whakamuri o \( f \), i tuhia \( f^{-1} \), he mahi e tutuki ana i:
\[
f^{-1}(f(x)) = x
\]
mō ia \( x \) i roto i te rohe o te mahi \( f \), me
\[
f(f^{-1}(y)) = y
\]
mō ia \( y \) i roto i te whānuitanga o te mahi \( f \).
Arā, ka whakakorea e te mahi whakamuri te mahi a te mahi taketake. Mena ka kiia a \( f \) he "tukanga," ko \( f^{-1} \) tōna tukanga whakamuri. Heoi, he mea nui kia whakanuia: ko te tohu \( f^{-1} \) ehara i te tikanga \( \frac{1}{f} \). He maha ngā wā ka hē te mārama o ngā ākonga ki tēnei. Ko te tohu e tohu ana i te whakamuri, ehara i te tauutuutu i roto i te tikanga haurua.
2. Rohe, Rohe-kotahi, me te Awhe o ngā Mahi Whakamuri
Kia mārama ai te ariā o te whakahurihanga, me mārama tātou ki te whanaungatanga i waenga i ngā huinga i roto i ngā mahi.
– Rohe: te huinga o ngā tāuru katoa ka taea te whakauru ki te mahi \(f\).
– Rohe-kotahi: te huinga o ngā putanga ūnga e ai ki te whakamāramatanga mahi.
– Awhe (horahanga hua): te huinga o ngā putanga e puta mai ana i te rohe.
Mō te mahi whakamuri, he whakahurihanga tūranga tēnei:
– Ko te rohe o \( f^{-1} \) ko te whānuitanga o \( f \) .
– Ko te whānuitanga o \( f^{-1} \) ko te rohe o \( f \) .
Koinei te take kāore he whakahurihanga o ngā mahi katoa: ki te kore te putanga o te mahi e "ahurei" e pā ana ki te tāuru, kāore e taea te whakatau whakahurihanga motuhake.
3. Ngā Tikanga mō te Whai Whakamuri o tētahi Mahi
He mahi whakamuri tō te mahi \( f \) (he mahi anō hoki) mēnā he mahi takirua a \( f \), arā:
1. Whakapūtātanga (takitahi-ki-tetahi): ka puta he putanga rerekē i ia tāuru rerekē.
I roto i te tikanga, mēnā ko \( f(a)=f(b) \) ko \( a=b \).
2. Tirohanga Matatau (ki runga): ka maherehia ia huānga o te rohe-ko e te rohe.
Ko te tikanga o tēnei he rite te awhe ki te kodomaine.
I roto i ngā horopaki kura, ko te aro nui ko te āhuatanga werohanga o ngā whakahuri hei mahi. Mena ehara te mahi i te werohanga, ka puta mai he putanga kotahi mai i ngā tāurunga rerekē e rua, nō reira kāore te "whakahurihanga" e whakaputa i tētahi uara ahurei.
Whakamātautau Raina Whakapae
Mō ngā mahi ka taea te tuhi kauwhata, he huarahi whai hua hei tirotiro i te werohanga: te whakamātautau rārangi whakapae.
Mena ka honoa te kauwhata e ia rārangi whakapae ki te pūwāhi kotahi, ko te mahi he kotahi-ki-te-tahi, ā, he tūponotanga kei a ia he whakamuri.
4. Me pēhea te whakatau i te mahi whakamuri
Ko ngā mahi whānui mō te kimi i te whakahuri o tētahi mahi taurangi ko:
1. Tuhia \( y = f(x) \).
2. Whakawhitihia ngā tūranga o \( x \) me \( y \): hangaia te \( x \) hei mahi a \( y \).
3. Whakaotia te whārite kia whiwhi ai i te \( y \).
4. Ko te hua whakamutunga ko \( y = f^{-1}(x) \).
Me titiro tātou ki tētahi tauira.
Tauira 1: Pānga Raina
Hei tauira \( f(x)=2x+3 \).
Hipanga:
1. \( y = 2x+3 \)
2. Whakawhiti: \( x = 2y+3 \)
3. Whakaotia: \( x-3 = 2y \Rightarrow y = \frac{x-3}{2} \)
4. Nō reira \( f^{-1}(x)=\frac{x-3}{2} \)
Ka taea e mātou te tirotiro:
\[
f(f^{-1}(x)) = 2\left(\frac{x-3}{2}\right)+3 = x-3+3=x
\]
Ko tōna tikanga he pono.
Tauira 2: Mahi Tapawhā (Me Here te Rohe)
Hei tauira, \( f(x)=x^2 \). He whakamuri tōna?
Ko te raruraru, \( f(2)=4 \) me \( f(-2)=4 \). Nō reira, ehara i te mea he werohanga puta noa i ngā tau tūturu katoa. Hei whai i tētahi whakahurihanga, me herea te rohe, hei tauira \( x \ge 0 \).
Mena ko te rohe ko \( [0,\infty) \), ko te whakahurihanga ko:
\[
f^{-1}(x) = \sqrt{x}
\]
Mena ko te rohe ko \( (-\infty,0] \), ko te whakahurihanga ko:
\[
f^{-1}(x) = -\sqrt{x}
\]
E whakaatu ana tēnei i te hiranga o te rohe i roto i ngā mahi whakamuri.
Tauira 3: Ngā Mahi Whaitake Māmā
Hei tauira \( f(x)=\frac{x-1}{x+2} \) me te tikanga \( x \ne -2 \).
1. \( y=\frac{x-1}{x+2} \)
2. Whakawhiti: \( x=\frac{y-1}{y+2} \)
3. Whakatauhia te \( y \):
\( x(y+2)=y-1 \Pere Matau xy+2x=y-1 \Pere Matau xy-y = -1-2x \Pere Matau y(x-1)=-(1+2x) \Pere Matau y=\frac{-(1+2x)}{x-1} \)
4. Nā reira:
\[
f^{-1}(x)=\frac{-(1+2x)}{x-1}
\]
Kia mahara ko \( x \ne 1 \) (nā te mea koinei te pūwāhi e kore ai te taupū i te whakamuri).
5. Te Hononga i waenga i ngā Kauwhata Mahi me ngā Whakamuri
I runga i te āhua ā-ira, ko ngā kauwhata o \( y=f(x) \) me \( y=f^{-1}(x) \) he whakaata i a rāua anō e pā ana ki te rārangi \( y=x \). Nā te mea i te whakahurihanga, ka huri te takirua raupapa \((x,y)\) hei \((y,x)\).
Hei tauira, mēnā kei runga i te kauwhata te pūwāhi \((1,5)\) \( y=f(x) \), kāti kei runga i te kauwhata te pūwāhi \((5,1)\) \( y=f^{-1}(x) \).
Mā tēnei māramatanga ka māmā ake te tirotiro ā-kanohi i ngā hua whakamuri, inā koa mō ngā mahi māmā noa iho.
6. Te Hanganga o te Mahi me te Tuakiri
He hononga tata ngā whakahurihuri ki te hanganga mahi. Mena he whakahurihuri kei a \( f \), kāti:
\[
(f \circ f^{-1})(x) = x \quad \text{and} \quad (f^{-1} \circ f)(x) = x
\]
ko te tikanga mā te hanganga o ngā mea e rua ka puta te mahi tuakiri, arā, he mahi e whakahoki mai ana i te tāuru e rite ana ki tōna āhua o nāianei.
Heoi, kia mōhio koe me ōrite ngā rohe. Hei tauira, ka mau te \( f^{-1}(f(x)) \) mō \( x \) i roto i te rohe o \( f \), ko te \( f(f^{-1}(x)) \) ka mau mō \( x \) i roto i te rohe o \( f^{-1} \) (arā, te whānuitanga o \( f \)).
7. Te Whakamahinga o ngā Mahi Whakamuri
Ehara i te mea he ariā whakarāpopoto noa iho te mahi whakamuri, engari he whānuitia te whakamahinga i roto i ngā mara maha:
1. Te whakaoti rapanga: Mena kei a tātou te \( y=f(x) \) ā, e hiahia ana tātou ki te kimi i te \( x \) mai i te uara o \( y \), ka whakamahia e tātou te whakahuri.
2. Te tahuritanga o ngā waeine me ngā tauine: Hei tauira, ko te tahuritanga o te pāmahana Celsius ki Fahrenheit, me te hurihanga hoki he takirua o ngā mahi whakamuri.
3. Te whakamunatanga māmā: He maha ngā wā ka hurihia ngā tukanga whakamunatanga me te wetewete (whakaaro whakamuri).
4. Tauira pūtaiao: He maha ngā tātai ahupūngao ka taea te huri, hei tauira mai i \( s=vt \) ka whiwhi tātou i te \( v=\frac{s}{t} \) me te \( t=\frac{s}{v} \) rānei i raro i ētahi tikanga.
8. Ngā Hapa Noa hei Ārai
Ko ētahi o ngā hapa noa ko:
– Ki te whakaaro ko \( f^{-1}(x) \) he rite ki \( \frac{1}{f(x)} \).
– I wareware ki te tuhi, ki te tirotiro rānei i te rohe, me te whakarite kia kaua te taupū e kore.
– Te kore e aro ki te mea me kotahi-ki-te-tahi tētahi mahi kia noho ai tōna whakamuri hei mahi anō hoki.
– Kāore e manatoko i ngā hua me te tito \( f(f^{-1}(x)) \) me \( f^{-1}(f(x)) \).
Te Katinga
Ko te mahi whakahurihuri he ariā e whakamārama ana i te āhua o te hurihanga o tētahi mahere kia hoki ai te putanga ki tōna urunga taketake. Heoi, kāore i te whai whakahurihuri ngā mahi katoa; ko te whakaritenga matua ko te mahi kia rua-whakaaro (kia werohia rānei ki runga i tētahi rohe motuhake). Mā te mārama ki te kimi i ngā whakahurihuri, ngā whanaungatanga awhe-rohe, ngā āhuatanga hanganga, me te whakamārama i ā rātou kauwhata, ka pai ake tā tātou whakarite mō ngā raruraru taurangi me ngā tono o te ao tūturu. Mā te mōhio ki ngā kaupapa matua o ngā mahi whakahurihuri ka whakaratohia he whakaritenga nui mō ngā kaupapa pāngarau matatau ake, pērā i te logarithms (te whakahurihuri o ngā taupū), te trigonometry whakahurihuri, me te tātaitai.
