Ihe atụ nke Ajụjụ na Mkparịta ụka nke Ọrụ Aljebra
Ọrụ algebra bụ isiokwu dị oke mkpa na mgbakọ na mwepụ, nke na-apụtakarị na ule ụlọ akwụkwọ na asọmpi mgbakọ na mwepụ. Ịghọta echiche nke ọrụ algebra na otu esi edozi nsogbu ndị metụtara ya bụ isi ihe dị mkpa iji mụta isiokwu a. Isiokwu a ga-akọwapụta ọtụtụ nsogbu atụ ma tụlee ọrụ algebra nke ọma.
Pendahuluan
Ọrụ bụ mmekọrịta nke na-ejikọta ihe ọ bụla dị n'otu setịpụ (a na-akpọ domain) na otu ihe kpọmkwem dị na setịpụ ọzọ (a na-akpọ codomain). N'ime mgbakọ na mwepụ, enwere ike ịkọwa ọrụ dị ka \( f : A \to B \), ebe \( f \) bụ ọrụ nke na-egosi ihe dị na setịpụ \( A \) na ihe dị na setịpụ \( B \ \). Ihe ndekọ izugbe maka ọrụ bụ \( f(x) \), nke pụtara na \( f \) bụ ọrụ nke dabere na mgbanwe \( x \).
Ajụjụ Ihe atụ nke 1: Ọrụ Linear
Ajụjụ: Chọpụta nha nhata nke ahịrị \( f(x) \) nke gafere isi ihe (2, 3) ma nwee gradient nke 4.
Azịza:
Ọrụ linear izugbe nwere ụdị \( f(x) = mx + c \), ebe \( m \) bụ gradient na \( c \) bụ y-intercept.
1. Tinye uru gradient \( m = 4 \) n'ime usoro nha anya:
\[
f(x) = 4x + c
\]
2. Jiri isi ihe (2, 3) chọta \( c \):
\[
3 = 4(2) + c
\]
\[
3 = 8 + c
\]
\[
c = 3 – 8
\]
\[
c = -5
\]
3. Site na \( m = 4 \) na \( c = -5 \), nha nha ahịrị ahụ bụ:
\[
f(x) = 4x – 5
\]
Ajụjụ Ihe atụ nke 2: Ọrụ Quadratic
Ajụjụ: E nyere ọrụ quadratic \( f(x) = ax^2 + bx + c \). Ọ bụrụ na eserese nke ọrụ ahụ gafere isi ihe (1, 4), (2, 7), na (3, 12), chọpụta uru nke \(a \), \( b \), na \( c \).
Azịza:
1. Tinye isi ihe (1, 4) n'ime usoro nhazi ahụ:
\[
4 = a(1)^2 + b(1) + c
\]
\[
4 = a + b + c \quad \text{(Nhazi 1)}
\]
2. Tinye isi ihe (2, 7) n'ime usoro nhazi ahụ:
\[
7 = a(2)^2 + b(2) + c
\]
\[
7 = 4a + 2b + c \quad \text{(Nhazi 2)}
\]
3. Tinye isi ihe (3, 12) n'ime usoro nhazi ahụ:
\[
12 = a(3)^2 + b(3) + c
\]
\[
12 = 9a + 3b + c \quad \text{(Nhazi 3)}
\]
4. Dozie usoro nke nha nhata ahịrị:
– Wepụ nha anya 1 site na nha anya 2:
\[
(7 – 4) = (4a + 2b + c) – (a + b + c)
\]
\[
3 = 3a + b \quad \text{(Nhazi 4)}
\]
– Wepụ nha anya 2 site na nha anya 3:
\[
(12 - 7) = (9a + 3b + c) - (4a + 2b + c)
\]
\[
5 = 5a + b \quad \text{(Nhazi 5)}
\]
5. Wepụ Nha nhata 4 site na Nha nhata 5:
\[
(5 - 3) = (5a + b) - (3a + b)
\]
\[
2 = 2a
\]
\[
na = 1
\]
6. Tinye \( a = 1 \) n'ime Nha nhata nke 4:
\[
3 = 3(1) + b
\]
\[
3 = 3 + b
\]
\[
b = 0
\]
7. Tinye \( a = 1 \) na \( b = 0 \) n'ime Nha 1:
\[
4 = 1 + 0 + c
\]
\[
c = 3
\]
Ya mere, ụkpụrụ nke \(a \), \(b \), na \(c \) bụ:
\[
a = 1, \quad b = 0, \quad c = 3
\]
Ya mere, ọrụ quadratic bụ:
\[
f(x) = x^2 + 3
\]
Ajụjụ Ihe Nlereanya nke 3: Ọrụ na Trigonometry
Nsogbu: E nyere ọrụ \( f(x) = 2 \sin (x) + \cos (x) \). Chọpụta \( f\left(\frac{\pi}{2}\right) \).
Azịza:
1. Tinye \( x = \frac{\pi}{2} \) n'ime ọrụ ahụ:
\[
f\left(\frac{\pi}{2}\right) = 2 \sin \left(\frac{\pi}{2}\right) + \cos \left(\frac{\pi}{2}\right)
\]
2. Cheta na ụkpụrụ trigonometric:
\[
\sin\left(\frac{\pi}{2}\right) = 1 \quad \text{na} \quad \cos\left(\frac{\pi}{2}\right) = 0
\]
3. Mgbe ahụ anyị ga-enweta:
\[
f\left(\frac{\pi}{2}\right) = 2(1) + 0
\]
\[
f\left(\frac{\pi}{2}\nri) = 2
\]
Ajụjụ Ihe Nlereanya nke 4: Nhazi nke Ọrụ
Nsogbu: E nyere ọrụ \( f(x) = 2x + 1 \) na \( g(x) = x^2 – 3 \). Chọpụta \( (f \circ g)(x) \) na \( (g \circ f)(x) \).
Azịza:
1. \((f \circ g)(x) \):
\[
(f \circ g)(x) = f(g(x))
\]
Tinye \( g(x) \) n'ime \( f(x) \):
\[
g(x) = x^2 – 3
\]
\[
f(g(x)) = f(x^2 – 3)
\]
Tinye \( f(x) = 2x + 1 \):
\[
f(x^2 – 3) = 2(x^2 – 3) + 1
\]
\[
= 2x^2 – 6 + 1
\]
\[
= 2x^2 – 5
\]
2. \((g \circ f)(x) \):
\[
(g \circ f)(x) = g(f(x))
\]
Tinye \( f(x) \) n'ime \( g(x) \):
\[
f(x) = 2x + 1
\]
\[
g(f(x)) = g(2x + 1)
\]
Tinye \( g(x) = x^2 – 3 \):
\[
g(2x + 1) = (2x + 1)^2 – 3
\]
\[
= 4x^2 + 4x + 1 – 3
\]
\[
= 4x^2 + 4x – 2
\]
Ya mere, nsonaazụ ikpeazụ:
\[
(f \circ g)(x) = 2x^2 – 5
\]
\[
(g \circ f)(x) = 4x^2 + 4x – 2
\]
Mmechi
Ọrụ algebraị gụnyere ọtụtụ akụkụ, site na ọrụ linear ruo ọrụ quadratic ruo na nhazi ọrụ. Isiokwu a na-egosi ọtụtụ nsogbu atụ yana mkparịta ụka zuru ezu. Ịghọta otu esi edozi nsogbu ndị a ga-abara uru n'ịmụta isiokwu nke ọrụ algebra na itinye echiche mgbakọ na mwepụ ndị ọzọ n'ọrụ.
Site na omume mgbe niile na nghọta siri ike nke echiche ndị a, idozi nsogbu ọrụ algebra ga-aghọ nkà a pụrụ ịtụkwasị obi. Nọgide na-eme ihe ma egbula oge ịchọ ihe ndị ọzọ iji mee ka nghọta gị dịkwuo omimi.
