Te Hanganga o ngā Mahi me ngā Mahi Whakamuri
I roto i te pāngarau, he taputapu tino noa ngā mahi hei whakaahua i ngā whanaungatanga i waenga i ngā huinga e rua. I roto i tēnei tuhinga, ka matapakihia e mātou ngā ariā nui e rua i roto i te ariā mahi: te hanganga mahi me ngā mahi whakamuri. He whānuitia ngā tono o ēnei e rua i roto i ngā peka pūtaiao, tae atu ki te pāngarau, te ahupūngao, te ōhanga, me te pūtaiao rorohiko.
1. Te Mārama ki ngā Mahi
I mua i tā tātou ruku hohonu ki te kaupapa o te hanganga mahi me te hurihanga, me mārama tātou ki te tikanga o te mahi. Ko te mahi he ture e hono ana i ia huānga o tētahi huinga, e kiia nei ko te rohe, ki tētahi huānga kotahi i tētahi atu huinga, e kiia nei ko te rohe-kotahi. Mena he mahi \( f \) e hono ana i tētahi huānga \( x \) o te rohe \( X \) ki tētahi huānga \( y \) o te rohe-kotahi \( Y \), kātahi ka tuhia \( f : X \rightarrow Y \) me \( y = f(x) \).
2. Te Hanganga o te Mahi
Ko te hanganga mahi he mahi pāngarau e tango ana i ngā mahi e rua \( f \) me \( g \) ka puta he mahi tuatoru, koinei te hua o te whakamahi i te \( f \) i muri i te \( g \). I roto i te tikanga, mēnā ko \( f : A \rightarrow B \) me \( g : B \rightarrow C \), ko te hanganga o te mahi \( g \) i muri i te \( f \), i tuhia ko \( g \circ f \), he mahi mai i \( A \) ki \( C \). Mō ia \( x \) i roto i te \( A \), ko te hua o te mahi hanganga ko \( (g \circ f)(x) = g(f(x)) \).
Tauira o te Hanganga Mahi
Me titiro tātou ki tētahi tauira tuturu hei mārama ake i te ariā o te hanganga mahi. Me kī he rua ā tātou mahi penei:
1. \( f(x) = 2x + 3 \)
2. \( g(x) = x^2 \)
E hiahia ana mātou ki te kimi i te uara o \( (g \circ f)(x) \). Mā te whakamāramatanga o te hanganga mahi, ka whakamahia tuatahitia e mātou te mahi \( f \) ki \( x \), kātahi ka whakamahia te mahi \( g \) ki te hua.
– \( f(x) = 2x + 3 \)
– \( g(f(x)) = g(2x + 3) = (2x + 3)^2 \)
Nō reira, \( (g \circ f)(x) = (2x + 3)^2 \).
Ngā Āhuatanga o te Hanganga Mahi
He maha ngā āhuatanga whakamere o te hanganga mahi e whakamahia whānuitia ana i roto i te tātari pāngarau:
1. Honohono: He mahi hono te hanganga mahi, arā, mēnā he mahi ōrite a \( f, g, \) me \( h \), kāti ko \( h \circ (g \circ f) = (h \circ g) \circ f \).
2. Tuakiri Hanganga: Mena he mahi tuakiri \( I \) ko ia huānga ko ia anō, kāti mō ia mahi \( f \), e mau ana ko \( f \circ I = I \circ f = f \).
3. Pānga Whakamuri
Ko te mahi whakamuri he mahi e "whakahuri" ana i te pānga o te mahi taketake. Mena ka hono te mahi \( f \) i ngā huānga \( x \) i roto i te rohe ki ngā huānga \( y \) i roto i te rohe-ko, ka hono te mahi whakamuri \( f^{-1} \) i \( y \) ki \( x \). Me noho takirua te mahi \( f \) (kotahi-ki-kotahi me te ki runga) kia whai whakamuri.
I roto i te tikanga, mēnā he mahi takirua a \( f: X \rightarrow Y \), ko te mahi whakamuri \( f^{-1}: Y \rightarrow X \) ka tautuhia e te āhuatanga e whai ake nei: \( f(f^{-1}(y)) = y \) mō ia \( y \) i roto i te \( Y \) me te \( f^{-1}(f(x)) = x \) mō ia \( x \) i roto i te \( X \).
Ngā Tauira o ngā Mahi Whakamuri
Whakaarohia te mahi \( f \) kua tautuhia ko \( f(x) = 2x + 3 \). Hei kimi i te mahi whakamuri \( f^{-1} \), me whakaoti e tātou te whārite \( y = 2x + 3 \) mō \( x \).
Ngā kaupae:
1. \( y = 2x + 3 \)
2. \( y – 3 = 2x \)
3. \( x = \frac{y – 3}{2} \)
Nō reira, ko te mahi whakamuri ko \( f^{-1}(y) = \frac{y – 3}{2} \).
Ngā Āhuatanga o ngā Mahi Whakamuri
Ko ētahi o ngā āhuatanga nui o ngā mahi whakamuri ko:
1. Takirua: Ko te whakahuri o te whakahuri ko te mahi taketake, arā, \( (f^{-1})^{-1} = f \).
2. Te Hanganga: Mō tētahi mahi takirua \( f \) me \( g \), ko te whakahuri o te hanganga ko te hanganga o ngā whakahuri i te raupapa whakamuri, arā, \( (g \circ f)^{-1} = f^{-1} \circ g^{-1} \).
3. Ngā Tuakiri: \( f^{-1}(f(x)) = x \) me \( f(f^{-1}(y)) = y \).
4. Te Whakamahinga o te Hanganga Mahi me ngā Mahi Whakamuri
He mea nui te hanganga mahi me ngā mahi whakamuri i roto i ngā tono mahi me ngā tono ariā maha. Anei ētahi tauira:
a. Tātaitai
I roto i te tātaitai, ka whakamahia te hanganga o ngā mahi ina whakamahia te ture mekameka mō te wehewehe. Mena ko \( y = g(u) \) me \( u = f(x) \), ko te pānga o \( y \) e pā ana ki \( x \) mā te whakamahi i te ture mekameka ko \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).
b. Te Whakamunatanga
I roto i te whakamunatanga hou, ka whakamahia ngā mahi whakahurihuri i roto i ngā rauropi wetemunatanga. He maha ngā wā ko te kī wetemunatanga te whakahurihuri o te kī whakamunatanga, e āhei ai te whakahoki i ngā raraunga kua whakamunatia ki tōna āhua taketake mā te whakamahi i te rauropi whakahurihuri.
c. Pūnaha Hihiri
I roto i te tātari pūnaha hihiri, ka whakamahia ngā mahi hei whakaahua i te whanaketanga o tētahi pūnaha i roto i te wā. Mā te mōhio ki te mahi whakamuri ka āwhina i te whakatau i te āhua tīmatanga o te pūnaha mēnā e mōhiotia ana te āhua whakamutunga.
5. Whakamutunga
Ko te hanganga mahi me ngā mahi whakamuri he ariā matua e rua i roto i te pāngarau e whānuitia ana te whakamahinga i roto i ngā mara maha. Mā te hanganga mahi ka taea e tātou te whakakotahi i ngā mahi e rua hei mahi kotahi, ko ngā mahi whakamuri ia ka taea e tātou te huri i te pānga o tētahi mahi. Mā te mārama ki ō rātou āhuatanga me ngā whakamahinga, ka taea e tātou te whakaoti i ngā momo raruraru uaua i roto i te pāngarau me ētahi atu pūtaiao tono.
Mā te mārama ki ēnei ariā e rua, ka taea e ngā kaipūtaiao me ngā kaihangarau te waihanga i ngā tauira me ngā otinga whai hua ake mō ngā raruraru e pā ana ki ā rātou ake mara.
