Nā nīnau hoʻohālike e kūkākūkā ana i ka hoʻonui ʻana a me ka mahele ʻana o nā hana
I ka makemakika, he pilina ka hana e hoʻopili ana i kēlā me kēia mea o kekahi seti i hoʻokahi mea pono i loko o kekahi seti ʻē aʻe. Hoʻike pinepine ʻia kahi hana ma ke ʻano he \( f(x) \), ʻo ia hoʻi, ʻo \( f \) kahi hana o \( x \). ʻO kekahi o nā hana e hiki ke hana ʻia me nā hana, ʻo ia ka hoʻonui a me ka mahele ʻana. Ma kēia ʻatikala, e nānā mākou i kekahi mau pilikia hoʻohālike a kūkākūkā i nā hana hoʻonui a me ka mahele ʻana o nā hana.
Hoʻonui Hana
ʻO ka hoʻonui ʻana i ka hana he hana kahi e hoʻonui ai mākou i ʻelua mau hana a ʻo ka hopena he hana hou. Manaʻo mākou he ʻelua mau hana kā mākou \( f(x) \) a me \( g(x) \). Hiki ke hōʻike ʻia ka huahana o kēia mau hana ʻelua e like me \( (f \cdot g)(x) \) a i ʻole \( f(x) \cdot g(x) \).
Laʻana Nīnau 1:
Hāʻawi ʻia i ʻelua mau hana:
– \( f(x) = 2x + 3 \)
– \( g(x) = x^2 – 4 \)
E huli i ka hopena o \( f(x) \cdot g(x) \).
Kūkākūkā:
ʻO ka huahana o kēia mau hana ʻelua:
\[ (f \cdot g)(x) = f(x) \cdot g(x) \]
No laila:
\[ (f \cdot g)(x) = (2x + 3) \cdot (x^2 – 4) \]
No ka hoʻonui ʻana i ʻelua polynomials, hoʻohana mākou i ke ʻano distributive:
\[ (2x + 3)(x^2 – 4) = 2x(x^2) + 2x(-4) + 3(x^2) + 3(-4) \]
\[ = 2x^3 – 8x + 3x^2 – 12 \]
No laila, ʻo ka hopena hope loa:
\[ (f \cdot g)(x) = 2x^3 + 3x^2 – 8x – 12 \]
Laʻana Nīnau 2:
Hana i hāʻawi ʻia:
– \( f(x) = \sin(x) \)
– \( g(x) = \cos(x) \)
E huli i ka hopena o \( f(x) \cdot g(x) \).
Kūkākūkā:
ʻO ka huahana o kēia mau hana ʻelua:
\[ (f \cdot g)(x) = \sin(x) \cdot \cos(x) \]
No laila, ʻo ka hopena hope loa:
\[ (f \cdot g)(x) = \sin(x) \cos(x) \]
Ma ka trigonometry, ʻike mākou:
\[ \sin(x) \cos(x) = \frac{1}{2} (\sin(2x)) \]
No laila, ʻo ka hopena o ka hoʻonui ʻana i kēia mau hana:
\[ (f \cdot g)(x) = \frac{1}{2} \sin(2x) \]
Māhele o nā Hana
ʻO ka mahele hana ka hana kahi e puʻunaue ai mākou i kekahi hana i kekahi a ʻo ka hopena he hana hou, inā ʻaʻole like ka mahele me ka ʻole. Manaʻo mākou he ʻelua mau hana kā mākou \( f(x) \) a me \( g(x) \). Hiki ke hōʻike ʻia ka mahele ʻana o kēia mau hana ʻelua e like me \( \left( \frac{f}{g} \right)(x) \) a i ʻole \( \frac{f(x)}{g(x)} \).
Laʻana Nīnau 3:
Hāʻawi ʻia i ʻelua mau hana:
– \( f(x) = x^2 – 1 \)
– \( g(x) = x – 1 \)
E huli i ka hopena o \( \frac{f(x)}{g(x)} \).
Kūkākūkā:
ʻO ka mahele ʻana o kēia mau hana ʻelua penei:
\[ \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \]
No laila:
\[ \left( \frac{f}{g} \right)(x) = \frac{x^2 – 1}{x – 1} \]
Hiki iā kākou ke hoʻomaʻalahi i nā hakina ma ka hoʻohui ʻana i ka helu:
\[ x^2 – 1 = (x + 1)(x – 1) \]
No laila:
\[ \left( \frac{f}{g} \right)(x) = \frac{(x + 1)(x – 1)}{x – 1} \]
No ka mea, hiki iā mākou ke hoʻopau iā \( x \neq 1 \), ma ka numerator a me ka denominator, hiki iā mākou ke hoʻopau iā \( (x – 1) \):
\[ \left( \frac{f}{g} \right)(x) = x + 1 \]
Laʻana Nīnau 4:
Hāʻawi ʻia i ʻelua mau hana:
– \( f(x) = e^x \)
– \( g(x) = x \)
E huli i ka hopena o \( \frac{f(x)}{g(x)} \).
Kūkākūkā:
ʻO ka mahele ʻana o kēia mau hana ʻelua penei:
\[ \left( \frac{f}{g} \right)(x) = \frac{e^x}{x} \]
No laila, ʻo ka hopena hope loa:
\[ \left( \frac{f}{g} \right)(x) = \frac{e^x}{x} \]
Laʻana Nīnau 5:
Hana i hāʻawi ʻia:
– \( f(x) = \ln(x) \)
– \( g(x) = x^2 \)
E huli i ka hopena o \( \frac{f(x)}{g(x)} \).
Kūkākūkā:
ʻO ka mahele ʻana o kēia mau hana ʻelua penei:
\[ \left( \frac{f}{g} \right)(x) = \frac{\ln(x)}{x^2} \]
No laila, ʻo ka hopena hope loa:
\[ \left( \frac{f}{g} \right)(x) = \frac{\ln(x)}{x^2} \]
Ka hopena
ʻO ka hoʻonui a me ka puʻunaue ʻana o nā hana he mau manaʻo koʻikoʻi ia i ka makemakika a he mea pono loa ia i nā ʻano noi like ʻole, ma ka makemakika maʻemaʻe a me nā ʻepekema i hoʻopili ʻia e like me ke kino a me ka ʻenekinia. Ma ka hoʻomaopopo ʻana i ke ʻano o ka hoʻonui a me ka puʻunaue ʻana i nā hana, hiki iā mākou ke hoʻoponopono i nā pilikia like ʻole e pili ana iā lākou. ʻO ke kūkākūkā ʻana i nā pilikia ma luna nei e hāʻawi i ka ʻike i ke ʻano o ka hana ʻana i kēia mau hana a me nā hopena i loaʻa.
E hoʻomau i ka hoʻomaʻamaʻa ʻana e hoʻonui i kou ʻike i kēia mea, ʻoiai he mea nui ka ʻike paʻa o nā hana hana no ka holomua ʻana i nā haʻawina makemakika hou aʻe. Inā loaʻa iā ʻoe kekahi mau pilikia, mai kānalua e nīnau i kāu kumu a i ʻole e ʻimi i nā kumuwaiwai aʻo hou aʻe. Lana ko mākou manaʻo ua kōkua kēia ʻatikala i ka hoʻomaopopo ʻana i ka hoʻonui a me ka mahele ʻana o nā hana.
