Nā Laʻana o nā Nīnau a me ke Kūkākūkā ʻana o nā Hana Algebraic
He kumuhana koʻikoʻi nā hana algebra i ka makemakika, e ʻike pinepine ʻia ana i nā hoʻokolohua kula a me nā hoʻokūkū makemakika. ʻO ka hoʻomaopopo ʻana i ke kumumanaʻo o nā hana algebra a pehea e hoʻoponopono ai i nā pilikia pili ke kī i ka hoʻopaʻa ʻana i kēia kumuhana. E hōʻike ana kēia ʻatikala i kekahi mau pilikia hoʻohālike a kūkākūkā i nā hana algebra me nā kikoʻī.
Pendahuluan
ʻO kahi hana he pilina e hoʻopili ana i kēlā me kēia mea i loko o hoʻokahi set (i kapa ʻia ʻo ke kikowaena) i hoʻokahi mea pono i loko o kekahi set ʻē aʻe (i kapa ʻia ʻo ke codomain). Ma ke ʻano makemakika, hiki ke hōʻike ʻia kahi hana e like me \( f : A \to B \), kahi \( f \) he hana e hoʻohālikelike ana i nā mea i loko o ka set \( A \) i nā mea i loko o ka set \( B \). ʻO ka hōʻailona laulā no kahi hana ʻo \( f(x) \), ʻo ia hoʻi, ʻo \( f \) he hana e hilinaʻi ana i ka loli \( x \).
Laʻana Nīnau 1: Hana Linear
Nīnau: E hoʻoholo i ka hoohalike o ka laina \( f(x) \) e hele ana ma ke kiko (2, 3) a he 4 kona gradient.
Kūkākūkā:
ʻO ke ʻano o ka hana linear laulā he \( f(x) = mx + c \), kahi ʻo \( m \) ka gradient a ʻo \( c \) ka y-intercept.
1. E hoʻololi i ka waiwai gradient \( m = 4 \) i loko o ka hoohalike:
\[
f(x) = 4x + c
\]
2. E hoʻohana i ke kiko (2, 3) e ʻike ai i ka \( c \):
\[
3 = 4(2) + c
\]
\[
3 = 8 + c
\]
\[
c = 3 – 8
\]
\[
c = -5
\]
3. Me \( m = 4 \) a me \( c = -5 \), ʻo ke kaulike o ka laina penei:
\[
f(x) = 4x – 5
\]
Laʻana Nīnau 2: Nā Hana Quadratic
Nīnau: Hāʻawi ʻia kahi hana quadratic \( f(x) = ax^2 + bx + c \). Inā hele ka pakuhi o ka hana ma o nā kiko (1, 4), (2, 7), a me (3, 12), e hoʻoholo i nā waiwai o \( a \), \( b \), a me \( c \).
Kūkākūkā:
1. E pani i ke kiko (1, 4) i loko o ka hoohalike:
\[
4 = a(1)^2 + b(1) + c
\]
\[
4 = a + b + c \quad \text{(Equation 1)}
\]
2. E pani i ke kiko (2, 7) i loko o ka hoohalike:
\[
7 = a(2)^2 + b(2) + c
\]
\[
7 = 4a + 2b + c \quad \text{(Kaulike 2)}
\]
3. E pani i ke kiko (3, 12) i loko o ka hoohalike:
\[
12 = a(3)^2 + b(3) + c
\]
\[
12 = 9a + 3b + c \quad \text{(Kaulike 3)}
\]
4. E hoʻoponopono i ka ʻōnaehana o nā kaulike linear:
– E unuhi i ka Hoʻohālikelike 1 mai ka Hoʻohālikelike 2:
\[
(7 – 4) = (4a + 2b + c) – (a + b + c)
\]
\[
3 = 3a + b \quad \text{(Equation 4)}
\]
– E unuhi i ka Hoʻohālikelike 2 mai ka Hoʻohālikelike 3:
\[
(12 – 7) = (9a + 3b + c) – (4a + 2b + c)
\]
\[
5 = 5a + b \quad \text{(Equation 5)}
\]
5. E unuhi i ka Hoʻohālikelike 4 mai ka Hoʻohālikelike 5:
\[
(5 – 3) = (5a + b) – (3a + b)
\]
\[
2 = 2a
\]
\[
i = 1
\]
6. E hoʻololi iā \( a = 1 \) i loko o ka Hoʻohālikelike 4:
\[
3 = 3(1) + b
\]
\[
3 = 3 + b
\]
\[
b = 0
\]
7. E pani iā \( a = 1 \) a me \( b = 0 \) i loko o ka Equation 1:
\[
4 = 1 + 0 + c
\]
\[
c = 3
\]
No laila, ʻo nā waiwai o \(a \), \(b \), a me \(c \):
\[
a = 1, \quad b = 0, \quad c = 3
\]
No laila, ʻo ka hana quadratic penei:
\[
f(x) = x^2 + 3
\]
Laʻana Nīnau 3: Nā Hana a me ka Trigonometry
Pilikia: Hāʻawi ʻia kahi hana \( f(x) = 2 \sin (x) + \cos (x) \). E hoʻoholo \( f\left(\frac{\pi}{2}\right) \).
Kūkākūkā:
1. E hoʻololi iā \( x = \frac{\pi}{2} \) i loko o ka hana:
\[
f\left(\frac{\pi}{2}\right) = 2 \sin \left(\frac{\pi}{2}\right) + \cos \left(\frac{\pi}{2}\right)
\]
2. E hoʻomanaʻo i nā waiwai trigonometric:
\[
\sin\left(\frac{\pi}{2}\right) = 1 \quad \text{and} \quad \cos\left(\frac{\pi}{2}\right) = 0
\]
3. A laila loaʻa iā mākou:
\[
f\left(\frac{\pi}{2}\right) = 2(1) + 0
\]
\[
f\left(\frac{\pi}{2}\right) = 2
\]
Laʻana Nīnau 4: Ka Hoʻonohonoho ʻana o nā Hana
Pilikia: Hāʻawi ʻia nā hana \( f(x) = 2x + 1 \) a me \( g(x) = x^2 – 3 \). E hoʻoholo \( (f \circ g)(x) \) a me \( (g \circ f)(x) \).
Kūkākūkā:
1. \( (f \circ g)(x) \) :
\[
(f \circ g)(x) = f(g(x))
\]
E hoʻololi iā \( g(x) \) i loko o \( f(x) \):
\[
g(x) = x^2 – 3
\]
\[
f(g(x)) = f(x^2 – 3)
\]
E hoʻopili \( f(x) = 2x + 1 \):
\[
f(x^2 – 3) = 2(x^2 – 3) + 1
\]
\[
= 2x^2 – 6 + 1
\]
\[
= 2x^2 – 5
\]
2. \( (g \circ f)(x) \) :
\[
(g \circ f)(x) = g(f(x))
\]
E hoʻololi iā \( f(x) \) i loko o \( g(x) \):
\[
f(x) = 2x + 1
\]
\[
g(f(x)) = g(2x + 1)
\]
E hoʻopili \( g(x) = x^2 – 3 \):
\[
g(2x + 1) = (2x + 1)^2 – 3
\]
\[
= 4x^2 + 4x + 1 – 3
\]
\[
= 4x^2 + 4x – 2
\]
No laila, ka hopena hope loa:
\[
(f \circ g)(x) = 2x^2 – 5
\]
\[
(g \circ f)(x) = 4x^2 + 4x – 2
\]
Ka hopena
Hoʻopili nā hana algebraic i nā ʻano he nui, mai nā hana linear a i nā hana quadratic a i nā haku mele hana. Hōʻike kēia ʻatikala i kekahi mau pilikia hoʻohālike me nā kūkākūkā kikoʻī. ʻO ka hoʻomaopopo ʻana i ke ʻano o ka hoʻoponopono ʻana i kēia mau pilikia e waiwai nui ia i ka hoʻopaʻa ʻana i ke kumuhana o nā hana algebraic a me ka hoʻopili ʻana o nā manaʻo makemakika ʻē aʻe.
Me ka hoʻomaʻamaʻa mau ʻana a me ka hoʻomaopopo pono ʻana i nā kumumanaʻo, e lilo ka hoʻoponopono ʻana i nā pilikia hana algebra i mākaukau hilinaʻi. E hoʻomau i ka hoʻomaʻamaʻa ʻana a mai kānalua e ʻimi i nā kumuwaiwai hou aʻe e hoʻoikaika i kou ʻike.
