Ajụjụ Ihe Nlereanya na Mkparịta ụka nke Nhazi Ọrụ
Nhazi ọrụ bụ echiche na mgbakọ na mwepụ ebe a na-ejikọta ọrụ abụọ n'otu. Ọ bụrụ na \(f \) na \(g \) bụ ọrụ abụọ, mgbe ahụ nhazi nke \(f \) na \(g \) bụ ọrụ ọhụrụ akọwapụtara dị ka \( (f \circ g)(x) \) nke pụtara \(f(g(x)) \). N'isiokwu a, anyị ga-atụle ọtụtụ nsogbu atụ na otu esi edozi ha metụtara nhazi ọrụ.
1. Nghọta Dị Mkpa nke Ihe mejupụtara Ọrụ
Tupu anyị abanye n'ajụjụ ihe atụ, ka anyị ghọta nke ọma ihe nhazi ọrụ bụ.
Ka e were ya na e nwere ọrụ abụọ \( f \) na \( g \):
– Ọrụ \( f \) : \( x \mapsto f(x) \)
– Ọrụ \( g \) : \( x \mapsto g(x) \)
Ihe mejupụtara \(f \) na \(g \), nke e dere dịka \(f \circ g \), bụ ọrụ nke na-emezu:
\[ (f \circ g)(x) = f(g(x)) \]
N'ebe a, \(g(x) \) bụ ntinye na ọrụ \(f \).
2. Ihe atụ Ajụjụ nke 1
Ajụjụ:
E nyere ọrụ \( f(x) = 2x + 3 \) na ọrụ \( g(x) = x – 5 \). Chọpụta \( (f \circ g)(x) \) na \( (g \circ f)(x) \).
Azịza:
Ka anyị gbakọọ ihe mejupụtara mbụ \((f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
Nzọụkwụ mbụ, anyị na-etinye \( g(x) \) n'ime \( f(x) \):
\[ g(x) = x – 5 \]
\[ f(g(x)) = f(x – 5) \]
Nzọụkwụ nke abụọ, anyị na-etinye \( x – 5 \) n'ime ọrụ \( f \):
\[ f(x – 5) = 2(x – 5) + 3 \]
\[ = 2x – 10 + 3 \]
\[ = 2x – 7 \]
Ya mere, \((f \circ g)(x) = 2x – 7 \).
Ugbua, ka anyị gbakọọ ihe mejupụtara nke abụọ \((g \circ f)(x) \):
\[ (g \circ f)(x) = g(f(x)) \]
Nzọụkwụ mbụ, anyị na-etinye \( f(x) \) n'ime \( g(x) \):
\[ f(x) = 2x + 3 \]
\[ g(f(x)) = g(2x + 3) \]
Nzọụkwụ nke abụọ, anyị na-etinye \( 2x + 3 \) n'ime ọrụ \( g \):
\[ g(2x + 3) = (2x + 3) – 5 \]
\[ = 2x + 3 – 5 \]
\[ = 2x – 2 \]
Ya mere, \((g \circ f)(x) = 2x – 2 \).
3. Ihe atụ Ajụjụ nke 2: Nhazi nke Ọrụ na Ọrụ Quadratic
Ajụjụ:
E nyere ọrụ \( f(x) = x^2 + 1 \) na ọrụ \( g(x) = 3x – 4 \). Chọpụta \( (f \circ g)(x) \) na \( (g \circ f)(x) \).
Azịza:
Ka anyị gbakọọ ihe mejupụtara mbụ \((f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
Nzọụkwụ mbụ, anyị na-etinye \( g(x) \) n'ime \( f(x) \):
\[ g(x) = 3x – 4 \]
\[ f(g(x)) = f(3x – 4) \]
Nzọụkwụ nke abụọ, anyị na-abanye \(3x – 4 \) n'ime ọrụ \(f \):
\[ f(3x – 4) = (3x – 4)^2 + 1 \]
\[ = (3x – 4)(3x – 4) + 1 \]
\[ = 9x^2 – 12x \cdot 2 + 16 + 1 \]
\[ = 9x^2 – 24x + 16 + 1 \]
\[ = 9x^2 – 24x + 17 \]
Ya mere, ((f \circ g)(x) = 9x^2 – 24x + 17 \).
Ugbua, ka anyị gbakọọ ihe mejupụtara nke abụọ \((g \circ f)(x) \):
\[ (g \circ f)(x) = g(f(x)) \]
Nzọụkwụ mbụ, anyị na-etinye \( f(x) \) n'ime \( g(x) \):
\[ f(x) = x^2 + 1 \]
\[ g(f(x)) = g(x^2 + 1) \]
Nzọụkwụ nke abụọ, anyị na-etinye \(x^2 + 1 \) n'ime ọrụ \(g \):
\[ g(x^2 + 1) = 3(x^2 + 1) – 4 \]
\[ = 3x^2 + 3 – 4 \]
\[ = 3x^2 – 1 \]
Ya mere, \((g \circ f)(x) = 3x^2 – 1 \).
4. Ihe atụ Ajụjụ nke 3: Nhazi nke Ọrụ Trigonometric
Ajụjụ:
E nyere ọrụ \( f(x) = \sin x \) na ọrụ \( g(x) = x^2 \). Chọpụta \( (f \circ g)(x) \) na \( (g \circ f)(x) \).
Azịza:
Ka anyị gbakọọ ihe mejupụtara mbụ \((f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
Nzọụkwụ mbụ, anyị na-etinye \( g(x) \) n'ime \( f(x) \):
\[ g(x) = x^2 \]
\[ f(g(x)) = f(x^2) \]
Nzọụkwụ nke abụọ, anyị na-etinye \(x^2 \) n'ime ọrụ \(f \):
\[ f(x^2) = \mmehie (x^2) \]
Ya mere, \((f \circ g)(x) = \sin (x^2) \).
Ugbua, ka anyị gbakọọ ihe mejupụtara nke abụọ \((g \circ f)(x) \):
\[ (g \circ f)(x) = g(f(x)) \]
Nzọụkwụ mbụ, anyị na-etinye \( f(x) \) n'ime \( g(x) \):
\[ f(x) = \mmehie x \]
\[g(f(x)) = g(\sin x) \]
Nzọụkwụ nke abụọ, anyị na-etinye \(\sin x \) n'ime ọrụ \( g \):
\[ g(\sin x) = (\sin x)^2 \]
\[ = \mmehie^2 x \]
Ya mere, \((g \circ f)(x) = \sin^2 x \).
Mmechi
Nhazi ọrụ bụ ụzọ isi jikọta ọrụ abụọ n'otu ọrụ. Site na ihe atụ ndị dị n'elu, anyị mụtara na usoro nhazi ọrụ gụnyere itinye otu ọrụ n'ọnọdụ ọzọ. Nsonaazụ ikpeazụ nke nhazi ọrụ dabere nke ukwuu n'usoro e si etinye ọrụ ndị ahụ n'ọnọdụ mbụ.
Ọ dị mkpa ịghọta na \((f \circ g)(x) \) abụghị otu ihe ahụ mgbe niile na \((g \circ f)(x) \), ọdịiche a nwekwara ike ịdị oke mkpa n'ọtụtụ ojiji nke mgbakọ na mwepụ na sayensị. Ya mere, ịghọta ihe ndị bụ isi na otu esi agbakọ ihe mejupụtara ọrụ dị oke mkpa maka onye ọ bụla na-amụ mgbakọ na mwepụ n'ọkwa dị n'etiti ma ọ bụ nke dị elu.
Enwere m olileanya na mkparịta ụka na ajụjụ atụ dị n'elu ga-abara uru ma nyere ndị na-agụ akwụkwọ aka ịghọta ihe mejupụtara ọrụ dị iche iche.
