Ngā Tauira Pātai me te Kōrero mō te Hanganga Mahi
Ko te hanganga mahi he ariā i roto i te pāngarau e honoa ai ngā mahi e rua hei mahi kotahi. Mena he mahi e rua a \( f \) me \( g \), ko te hanganga o \( f \) me \( g \) he mahi hou kua tautuhia ko \( (f \circ g)(x) \) ko te tikanga ko \( f(g(x)) \). I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira rapanga me pēhea te whakaoti i aua rapanga e pā ana ki te hanganga mahi.
1. Te Māramatanga Taketake mō te Hanganga Mahi
I mua i te urunga atu ki ngā tauira pātai, me mārama poto tātou he aha te hanganga mahi.
Me kī e rua ngā mahi \( f \) me \( g \):
– Mahi \( f \) : \( x \mapsto f(x) \)
– Mahi \( g \) : \( x \mapsto g(x) \)
Ko te hanganga o \( f \) me \( g \), i tuhia ko \( f \circ g \), he mahi e tutuki ana i:
\[ (f \circ g)(x) = f(g(x)) \]
I konei, ko \( g(x) \) te tāuru ki te mahi \( f \).
2. Tauira Pātai 1
Pātai:
I runga i te mahi \( f(x) = 2x + 3 \) me te mahi \( g(x) = x – 5 \). Whakatauhia \( (f \circ g)(x) \) me \( (g \circ f)(x) \).
Kōrero:
Me tatau te hanganga tuatahi \( (f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
Ko te taahiraa tuatahi, ka tāpirihia e mātou a \( g(x) \) ki roto i a \( f(x) \):
\[ g(x) = x – 5 \]
\[ f(g(x)) = f(x – 5) \]
Tuarua, ka tāuruhia e tātou te \( x – 5 \) ki te mahi \( f \):
\[ f(x – 5) = 2(x – 5) + 3 \]
\[ = 2x – 10 + 3 \]
\[ = 2x – 7 \]
Nō reira, \( (f \circ g)(x) = 2x – 7 \).
Nā, me tatau tātou i te hanganga tuarua \( (g \circ f)(x) \):
\[ (g \circ f)(x) = g(f(x)) \]
Tuatahi, ka tāpirihia e mātou te \( f(x) \) ki roto i te \( g(x) \):
\[ f(x) = 2x + 3 \]
\[ g(f(x)) = g(2x + 3) \]
Ko te taahiraa tuarua, ka tāpirihia e mātou te \( 2x + 3 \) ki te mahi \( g \):
\[ g(2x + 3) = (2x + 3) – 5 \]
\[ = 2x + 3 – 5 \]
\[ = 2x – 2 \]
Nō reira, \( (g \circ f)(x) = 2x – 2 \).
3. Tauira Pātai 2: Te Hanganga o ngā Mahi me ngā Mahi Tapawhā
Pātai:
I runga i te mahi \( f(x) = x^2 + 1 \) me te mahi \( g(x) = 3x – 4 \). Whakatauhia \( (f \circ g)(x) \) me \( (g \circ f)(x) \).
Kōrero:
Me tatau te hanganga tuatahi \( (f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
Ko te taahiraa tuatahi, ka tāpirihia e mātou a \( g(x) \) ki roto i a \( f(x) \):
\[ g(x) = 3x – 4 \]
\[ f(g(x)) = f(3x – 4) \]
Ko te taahiraa tuarua, ka tāuruhia e tātou te \( 3x – 4 \) ki te mahi \( f \):
\[ f(3x – 4) = (3x – 4)^2 + 1 \]
\[ = (3x – 4)(3x – 4) + 1 \]
\[ = 9x^2 – 12x \cdot 2 + 16 + 1 \]
\[ = 9x^2 – 24x + 16 + 1 \]
\[ = 9x^2 – 24x + 17 \]
Na, \( (f \circ g)(x) = 9x^2 – 24x + 17 \).
Nā, me tatau tātou i te hanganga tuarua \( (g \circ f)(x) \):
\[ (g \circ f)(x) = g(f(x)) \]
Tuatahi, ka tāpirihia e mātou te \( f(x) \) ki roto i te \( g(x) \):
\[ f(x) = x^2 + 1 \]
\[ g(f(x)) = g(x^2 + 1) \]
Ko te taahiraa tuarua, ka tāuruhia e tātou te \( x^2 + 1 \) ki te mahi \( g \):
\[ g(x^2 + 1) = 3(x^2 + 1) – 4 \]
\[ = 3x^2 + 3 – 4 \]
\[ = 3x^2 – 1 \]
Nō reira, \( (g \circ f)(x) = 3x^2 – 1 \).
4. Tauira Pātai 3: Te Hanganga o ngā Mahi Pāngatoru
Pātai:
I runga i te mahi \( f(x) = \sin x \) me te mahi \( g(x) = x^2 \). Whakatauhia \( (f \circ g)(x) \) me \( (g \circ f)(x) \).
Kōrero:
Me tatau te hanganga tuatahi \( (f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
Ko te taahiraa tuatahi, ka tāpirihia e mātou a \( g(x) \) ki roto i a \( f(x) \):
\[ g(x) = x^2 \]
\[ f(g(x)) = f(x^2) \]
Tuarua, ka tāuruhia e tātou te \( x^2 \) ki te mahi \( f \):
\[ f(x^2) = \hara (x^2) \]
Nō reira, \( (f \circ g)(x) = \sin (x^2) \).
Nā, me tatau tātou i te hanganga tuarua \( (g \circ f)(x) \):
\[ (g \circ f)(x) = g(f(x)) \]
Tuatahi, ka tāpirihia e mātou te \( f(x) \) ki roto i te \( g(x) \):
\[ f(x) = \sin x \]
\[ g(f(x)) = g(\sin x) \]
Ko te taahiraa tuarua, ka tāpirihia e mātou te \( \sin x \) ki te mahi \( g \):
\[ g(\sin x) = (\sin x)^2 \]
\[ = \sin^2 x \]
Nō reira, \( (g \circ f)(x) = \sin^2 x \).
Whakamutunga
Ko te hanganga mahi he huarahi e whakakotahi ai i ngā mahi e rua ki roto i te mahi kotahi. Mā roto i ngā tauira i runga ake nei, i ako tātou ko te tukanga hanganga mahi he whakakapi i tētahi mahi mō tētahi atu. Ko te hua whakamutunga o te hanganga mahi e whakawhirinaki ana ki te raupapa o te whakamahinga tuatahi o ngā mahi.
He mea nui kia mārama ko \( (f \circ g)(x) \) kāore i te rite tonu ki \( (g \circ f)(x) \), ā, ka tino nui pea tēnei rerekētanga i roto i ngā momo whakamahinga o te pāngarau me te pūtaiao. Nō reira, he mea tino nui te mārama ki ngā kaupapa matua me te tatau i te hanganga o ngā mahi mō te hunga e ako ana i te pāngarau i te taumata waenga, i te taumata matatau rānei.
Ko te tumanako ka whai hua ngā kōrero me ngā tauira pātai i runga ake nei, ā, ka āwhina i ngā kaipānui ki te mārama ki te hanganga o ngā mahi.
