Ngā Tauira Pātai e Matapaki ana i ngā Mahi me ā Rātou Whakatauira
Pendahuluan
I roto i te pāngarau, he mea nui te mahi a ngā mahi hei taputapu mō te whakatauira i ngā āhuatanga o te ao tūturu. Mā ngā mahi ka taea e tātou te mārama ki te pānga o tētahi taurangi ki tētahi atu i roto i ngā horopaki maha, tae atu ki te ōhanga, te ahupūngao, te koiora, me te pūtaiao rorohiko. Ka kapi tēnei tuhinga i ētahi tauira o ngā mahi me tā rātou whakatauira, me te whakarato hoki i ngā whakamārama taipitopito hei āwhina i a koe ki te mārama ki ngā ariā matua.
Mahi: Whakamāramatanga me ngā Ariā Taketake
I mua i te ruku ki ngā tauira, me arotake tātou i ētahi ariā taketake mō ngā mahi. Ka taea te tautuhi i tētahi mahi hei ture e hono ana i ia huānga i roto i tētahi huinga, e kiia nei ko te rohe, ki tētahi huānga tika i roto i tētahi atu huinga, e kiia nei ko te rohe-ko. I roto i te pāngarau, ko te mahi \( f \) ka whakaatuhia i te āhua \( f(x) \), ko \( x \) he huānga o te rohe, ā, ko \( f(x) \) he huānga o te rohe-ko.
Tuhituhinga Mahi
– \( y = f(x) \) : I konei, ko \( x \) te taurangi motuhake, ko \( y \) ia te taurangi whakawhirinaki.
– Rohe: Ko te huinga o ngā uara ka taea mō \( x \).
– Rohe-kotahi: Ko te huinga o ngā uara ka taea mō \( y \).
Tauira Pātai 1: Pānga Rārangi
Pātai
I te mea kua hoatu te mahi \( f(x) = 3x + 2 \). Whakatauhia ngā uara o \( f(5) \) me \( f(-3) \).
Kōrero
Hei kimi i te \( f(x) \) i tētahi uara motuhake, ka whakakapia taua uara ki roto i te mahi.
– Kimihia \( f(5) \)
\( f(x) = 3x + 2 \)
\( f(5) = 3(5) + 2 \)
\( f(5) = 15 + 2 \)
\( f(5) = 17 \)
– Kimihia \( f(-3) \)
\( f(x) = 3x + 2 \)
\( f(-3) = 3(-3) + 2 \)
\( f(-3) = -9 + 2 \)
\( f(-3) = -7 \)
Nō reira, \( f(5) = 17 \) me \( f(-3) = -7 \).
Tauira Pātai 2: Mahi Tapawhā
Pātai
Hoatu he mahi tapawhā \( g(x) = x^2 – 4x + 4 \). Whakatauhia te uara o \( g(2) \) me ngā pūtake o te mahi.
Kōrero
Ka tīmata tātou mā te tatau i te uara o \( g(2) \):
– Kimihia \( g(2) \)
\( g(x) = x^2 – 4x + 4 \)
\( g(2) = (2)^2 – 4(2) + 4 \)
\( g(2) = 4 – 8 + 4 \)
\( g(2) = 0 \)
Muri iho, ka kitea e tātou ngā pūtake o te mahi mā te kimi i te uara o \( x \) ina \( g(x) = 0 \).
– Te rapu i te Pūtake
\( x^2 – 4x + 4 = 0 \)
Tāpirihia ki te āhua \( (x-2)^2 = 0 \)
Nō reira, ko te pūtake ko \( x = 2 \) (pūtake māhanga).
Ko te uara o \( g(2) \) he 0, ā, ko tōna pūtake ko \( x = 2 \).
Tauira 3: Ngā Mahi Taupūnga
Pātai
I runga i te mahi taupūnga \( h(x) = 2^x \). Kimihia te uara o \( h(3) \), ā, whakatauhia mēnā kei te piki haere, kei te heke iho rānei \( h(x) \).
Kōrero
Mō tēnei mahi, ka tīmata mā te tatau i te uara o \( h(3) \):
– Kimihia \( h(3) \)
\( h(x) = 2^x \)
\( h(3) = 2^3 \)
\( h(3) = 8 \)
Kātahi ka tātarihia e mātou mēnā kei te piki haere, kei te heke iho rānei te mahi.
– Tātaritanga Monotonitanga
Nā te mea ko \( 2 > 1 \), he mahi taupūtanga piki haere te mahi \( 2^x \) , ko te tikanga ka piki haere te uara o \( h(x) \) i te pikinga ake o \( x \).
Ko te uara o \( h(3) \) he 8, ā, ko \( h(x) \) he mahi piki haere.
Tauira Pātai 4: Pānga Tauira
Pātai
I runga i te mahi taupūnga (rōkarithmic function) \( k(x) = \log_2 (x + 1) \). Kimihia te uara o \( k(7) \), ā, whakatauhia te rohe o te mahi.
Kōrero
Mō te mahi taupū, ka tīmata mā te kimi i te uara o \( k(7) \):
– Kimihia \( k(7) \)
\( k(x) = \log_2 (x + 1) \)
\( k(7) = \log_2 (7 + 1) \)
\( k(7) = \log_2 8 \)
\( k(7) = 3 \) (nā te mea \( 2^3 = 8 \))
Muri iho, ka kitea e tātou te rohe o te mahi.
– Te Rapu i ngā Rohe Whenua
Kia tautuhia ai te \( \log_2 (x + 1) \), me pai te tautohe o te logarithm:
\( x + 1 > 0 \)
\( x > -1 \)
Nō reira, ko te rohe o \( k(x) \) ko \( x > -1 \).
Ko te uara o \( k(7) \) he 3, ā, ko te rohe o te mahi \( k(x) \) he \( x > -1 \).
Te Katinga
Ko ngā mahi me tā rātou whakatauira he ariā matua i roto i te pāngarau e āhei ai tātou ki te whakaoti i te whānuitanga o ngā raruraru i roto i te pūtaiao me te oranga o ia rā. Mā te mārama ki te whakahaere me te tātari i ngā mahi, ka taea e tātou te whakaahua i ngā whanaungatanga i waenga i ngā taurangi rerekē me te hanga matapae i runga i ngā raraunga o nāianei. Kei roto i tēnei tuhinga ētahi tauira raruraru me ngā matapakinga mō ngā mahi rārangi, tapawhā, taupū, me te taupū, e tumanako ana mātou ka āwhina i a tātou ki te mārama ki te ariā o ngā mahi me ā rātou tono.
