Ngā Tauira Pātai e Matapaki ana i te Tāpiri me te Tango o ngā Mahi
He ariā taketake, he ariā nui hoki te tāpiri me te tango i ngā mahi pāngarau. Ehara i te mea he mea nui tēnei ariā i roto i ngā horopaki mātauranga anake, engari he maha hoki ngā whakamahinga mahi i roto i te oranga o ia rā me ētahi atu mara ako. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira o ngā rapanga tāpiri me te tango, me ngā whakamārama taipitopito.
Ngā Whakamāramatanga me ngā Ariā Taketake
I mua i te urunga atu ki ngā tauira pātai, me matapaki tātou i te whakamāramatanga me ngā ariā taketake o ngā mahi tāpiri me te tango.
Tāpiri Mahi
Mena e rua ā tātou mahi \( f(x) \) me \( g(x) \), ko te tapeke o ēnei mahi e rua he mahi hou e tautuhia ana penei:
\[ (f + g)(x) = f(x) + g(x) \]
Te Whakaiti i te Mahi
He rite tonu te whakamāramatanga o te tango mahi ki te tāpiri mahi. Mena e rua ā tātou mahi \( f(x) \) me \( g(x) \), ko te tango i ēnei mahi e rua he mahi hou kua whakamāramatia penei:
\[ (f – g)(x) = f(x) – g(x) \]
Ngā Pātai Tauira me te Kōrero
Me titiro tātou ki ētahi tauira rapanga hei whakamārama i tēnei ariā.
Tauira 1: Te Tāpiri i ngā Mahi Raina
Me kī ko te f(x) = 2x + 3 me te g(x) = x – 1. Whakatauhia (f + g)(x)).
Kōrero:
Ka taea e tātou te tāpiri i ngā mahi e rua mā te tāpiri i ngā kupu e rite ana.
\[
(f + g)(x) = f(x) + g(x)
\]
\[
(f + g)(x) = (2x + 3) + (x – 1)
\]
\[
(f + g)(x) = 2x + x + 3 – 1
\]
\[
(f + g)(x) = 3x + 2
\]
Nō reira, \( (f + g)(x) = 3x + 2 \).
Tauira 2: Te Tangohanga o ngā Mahi Raina
Me kī ko te f(x) = 4x + 5 me te g(x) = 2x – 3. Whakatauhia (f – g)(x)).
Kōrero:
Ka taea e tātou te whakaiti i ngā mahi e rua mā te tango i ngā kupu e rite ana.
\[
(f – g)(x) = f(x) – g(x)
\]
\[
(f – g)(x) = (4x + 5) – (2x – 3)
\]
\[
(f – g)(x) = 4x + 5 – 2x + 3
\]
\[
(f – g)(x) = 2x + 8
\]
Nō reira, \( (f – g)(x) = 2x + 8 \).
Tauira 3: Te Tāpiri i ngā Mahi Tapawhā
Me kī ko \( f(x) = x^2 + 2x + 1 \) me \( g(x) = -x^2 + 4x – 3 \). Whakatauhia \( (f + g)(x) \).
Kōrero:
Ka taea e tātou te tāpiri i ngā mahi e rua mā te tāpiri i ngā kupu e rite ana.
\[
(f + g)(x) = f(x) + g(x)
\]
\[
(f + g)(x) = (x^2 + 2x + 1) + (-x^2 + 4x – 3)
\]
\[
(f + g)(x) = x^2 – x^2 + 2x + 4x + 1 – 3
\]
\[
(f + g)(x) = 6x – 2
\]
Nō reira, \( (f + g)(x) = 6x – 2 \).
Tauira 4: Te Tango i ngā Mahi Tapawhā
Me kī ko te f(x) = 3x^2 – 2x + 4 \) me te g(x) = x^2 + x – 5 \). Whakatauhia \( (f – g)(x) \).
Kōrero:
Ka taea e tātou te whakaiti i ngā mahi e rua mā te tango i ngā kupu e rite ana.
\[
(f – g)(x) = f(x) – g(x)
\]
\[
(f – g)(x) = (3x^2 – 2x + 4) – (x^2 + x – 5)
\]
\[
(f – g)(x) = 3x^2 – x^2 – 2x – x + 4 + 5
\]
\[
(f – g)(x) = 2x^2 – 3x + 9
\]
Nō reira, \( (f – g)(x) = 2x^2 – 3x + 9 \).
Tauira 5: Te Tāpiri me te Tango i ngā Mahi Taupū
Me kī \( f(x) = e^x \) me \( g(x) = e^{-x} \). Whakatauhia:
1. \( (f + g)(x) \)
2. \( (f – g)(x) \)
Kōrero:
1. Mahi Tāpiri:
\[
(f + g)(x) = f(x) + g(x)
\]
\[
(f + g)(x) = e^x + e^{-x}
\]
Nō reira, \( (f + g)(x) = e^x + e^{-x} \).
2. Te Whakaiti i te Mahi:
\[
(f – g)(x) = f(x) – g(x)
\]
\[
(f – g)(x) = e^x – e^{-x}
\]
Nō reira, \( (f – g)(x) = e^x – e^{-x} \).
Tauira 6: Te Tāpiri me te Tango i ngā Mahi Pānga-toru
Me kī ko \( f(x) = \sin x \) me \( g(x) = \cos x \). Whakatauhia:
1. \( (f + g)(x) \)
2. \( (f – g)(x) \)
Kōrero:
1. Mahi Tāpiri:
\[
(f + g)(x) = f(x) + g(x)
\]
\[
(f + g)(x) = \sin x + \cos x
\]
Nō reira, \( (f + g)(x) = \sin x + \cos x \).
2. Te Whakaiti i te Mahi:
\[
(f – g)(x) = f(x) – g(x)
\]
\[
(f – g)(x) = \sin x – \cos x
\]
Nō reira, \( (f – g)(x) = \sin x – \cos x \).
Tauira 7: Te Whakamahinga o te Tāpiri me te Tango i ngā Mahi i roto i ngā Raru Ā-Tinana
Me kī e rua ngā mahi e whakaahua ana i te tūranga (i roto i ngā mita) o ngā waka e rua e haere ana i te ara kotahi i roto i te wā \( t \) (i roto i ngā hēkona).
Waka A: \( f(t) = 5t + 2 \)
Waka B: \( g(t) = 3t + 4 \)
Whakatauhia:
1. Te tūnga ngātahi o ngā motuka e rua.
2. Te rerekētanga o te tūranga o ngā waka e rua i te wā \( t \).
Kōrero:
1. Mahi Tāpiri:
\[
(f + g)(t) = f(t) + g(t)
\]
\[
(f + g)(t) = (5t + 2) + (3t + 4)
\]
\[
(f + g)(t) = 8t + 6
\]
Nō reira, ko te tūnga o ngā waka e rua i te wā \(t \) ko \(8t + 6 \) mita.
2. Te Whakaiti i te Mahi:
\[
(f – g)(t) = f(t) – g(t)
\]
\[
(f – g)(t) = (5t + 2) – (3t + 4)
\]
\[
(f – g)(t) = 2t – 2
\]
Nō reira, ko te rerekētanga o te tūranga o ngā waka e rua i te wā \(t \) he \(2t – 2 \) mita.
Whakamutunga
He mea tino nui te tāpiri me te tango i ngā mahi i roto i te pāngarau. Ka taea e tātou te tāpiri, te tango rānei i ngā mahi e rua mā te tāpiri, te tango rānei i ngā kupu e rite ana. Ehara i te mea he whai hua tēnei ariā i roto i te horopaki mātauranga anake, engari he maha hoki ngā tono mahi. Mā roto i ngā tauira pātai i runga ake nei, ko te tumanako ka mārama ake ngā kaipānui ki tēnei ariā.
