Ajụjụ Ihe Nlereanya Banyere Mmekọ Ọrụ na Ọrụ Ndị Na-agbanwe Agbanwe
Na mgbakọ na mwepụ, echiche nke nhazi ọrụ na ọrụ inverse bụ isiokwu abụọ jikọtara ọnụ nke dị oke mkpa maka nghọta dị elu dịka calculus, nyocha mgbakọ na mwepụ, na ozizi ọrụ. Isiokwu a ga-enyocha echiche abụọ ahụ site n'inye ọtụtụ ihe atụ na mkparịta ụka dị mfe nghọta. Ebumnuche bụ inyere ndị na-agụ akwụkwọ aka ịghọta otu nhazi ọrụ na inverse ọrụ si arụ ọrụ n'ụzọ bara uru karị.
1. Nhazi Ọrụ
Nhazi ọrụ bụ ọrụ nke ijikọta ọrụ abụọ ka ọ bụrụ otu. Ọ bụrụ na anyị nwere ọrụ abụọ \( f(x) \) na \( g(x) \), mgbe ahụ nhazi nke ọrụ ndị a bụ \( (f \circ g)(x) \), nke a na-agụ "f nhazi g nke x" ma ọ bụ "f nke g nke x." A kọwara nhazi a dị ka itinye ọrụ \( g(x) \) mbụ, wee tinye ọrụ \( f \) na nsonaazụ nke \( g(x) \).
Ajụjụ Ihe atụ nke 1:
E nyere ọrụ ndị dị na \( f(x) = 2x + 3 \) na \( g(x) = x^2 – 1 \). Chọta ihe mejupụtara \( (f \circ g)(x) \) na \( (g \circ f)(x) \).
Azịza:
1. Chọpụta \((f \circ g)(x) \):
\( (f \circ g)(x) = f(g(x)) \)
\( = f(x^2 – 1) \)
Tinye \( x^2 – 1 \) n'ime \( f(x) \):
\( f(x^2 – 1) = 2(x^2 – 1) + 3 \)
\( = 2x^2 – 2 + 3 \)
\( = 2x^2 + 1 \)
Ya mere, \((f \circ g)(x) = 2x^2 + 1 \).
2. Chọpụta \((g \circ f)(x) \):
\( (g \circ f)(x) = g(f(x)) \)
\( = g(2x + 3) \)
Tinye \( 2x + 3 \) n'ime \( g(x) \):
\( g(2x + 3) = (2x + 3)^2 – 1 \)
Jiri njirimara quadratic gbakọọ \( (2x + 3)^2 \):
\( = 4x^2 + 12x + 9 – 1 \)
\( = 4x^2 + 12x + 8 \)
Ya mere, \((g \circ f)(x) = 4x^2 + 12x + 8 \).
2. Ọrụ Mgbanwe
Ọrụ ntụgharị bụ ọrụ nke na-agbanwe mmetụta nke ọrụ mbụ ahụ. Ọ bụrụ na \(f \) bụ ọrụ, mgbe ahụ mgbanwe nke \(f \), nke e dere dị ka \(f^{-1} \), bụ ọrụ nke na-emezu \(f(f^{-1}(x)) = x \) na \(f^{-1}(f(x)) = x \).
Iji chọta ọrụ inverse nke ọrụ, anyị ga-eme ihe ndị a:
1. Jiri \( y \) dochie \( f(x) \) .
2. Dozie usoro maka \( x \) n'ihe gbasara \( y \).
3. Gbanwee mgbanwe \( x \) na \( y \).
Ajụjụ Ihe atụ nke 2:
Ọ bụrụ na e nyere ọrụ \( f(x) = 3x – 4 \), chọta ihe dị iche na ya, ya bụ \( f^{-1}(x) \).
Azịza:
1. Jiri \(y \) dochie \(f(x) \):
\( y = 3x – 4 \).
2. Dozie maka \( x \) n'ihe gbasara \( y \):
\( y = 3x – 4 \)
Tinye 4 n'akụkụ abụọ nke usoro nha anya ahụ:
\( y + 4 = 3x \)
Kewaa akụkụ abụọ nke nha nhata ahụ site na 3:
\( x = \frac{y + 4}{3} \)
3. Gbanwee mgbanwe \( x \) na \( y \):
\( f^{-1}(x) = \frac{x + 4}{3} \)
Ya mere, ihe dị iche nke \( f(x) = 3x – 4 \) bụ \( f^{-1}(x) = \frac{x + 4}{3} \).
3. Ajụjụ Ihe atụ nwere njikọta nke ihe mejupụtara na ntụgharị
Ajụjụ Ihe atụ nke 3:
E nyere ọrụ ndị dị na \( f(x) = x^3 + 2 \) na \( g(x) = \sqrt[3]{x – 2} \). Gosipụta na \( g(x) \) bụ ihe dị iche na \( f(x) \).
Azịza:
Iji gosi na \(g(x) \) bụ ihe dị iche na \(f(x) \), anyị ga-egosi na \((f \circ g)(x) = x \) na \((g \circ f)(x) = x \).
1. Gosi na \((f \circ g)(x) = x \):
\( (f \circ g)(x) = f(g(x)) \)
Tinye \( g(x) = \sqrt[3]{x – 2} \) n'ime \( f(x) \):
\( f(g(x)) = f(\sqrt[3]{x – 2}) \)
\( = (\sqrt[3]{x – 2})^3 + 2 \)
Maka na \(((\sqrt[3]{x – 2})^3 = x – 2 \):
\( = (x – 2) + 2 \)
\( = x \).
2. Gosi na \((g \circ f)(x) = x \):
\( (g \circ f)(x) = g(f(x)) \)
Dochie \( f(x) = x^3 + 2 \) n'ime \( g(x) \):
\( g(f(x)) = g(x^3 + 2) \)
\( = \sqrt[3]{(x^3 + 2) – 2} \)
\( = \sqrt[3]{x^3} \)
\( = x \).
Ebe ọ bụ na \((f \circ g)(x) = x \) na \((g \circ f)(x) = x \), mgbe ahụ \(g(x) \) bụ ihe dị iche na \(f(x) \).
4. Ngwa dị na Ndụ Kwa Ụbọchị
Ajụjụ Ihe atụ nke 4:
Ọkà mmụta sayensị na-eji ụdị mgbakọ na mwepụ abụọ nke ọrụ \( f(T) = 5T + 40 \) na \( g(P) = \frac{P – 40}{5} \), ebe \( T \) bụ okpomọkụ na Celsius na \( P \) bụ nrụgide dị na Pascals. Chọpụta ma ọrụ \( g \) ọ bụ mgbanwe nke ọrụ \( f \).
Azịza:
Iji gosi na \(g \) bụ ihe dị iche na \(f \), anyị ga-egosi na \((f \circ g)(P) = P \) na \((g \circ f)(T) = T \).
1. Gosi na \((f \circ g)(P) = P \):
\( (f \circ g)(P) = f(g(P)) \)
Dochie \( g(P) = \frac{P – 40}{5} \) n'ime \( f(T) \):
\( f(g(P)) = f\left(\frac{P – 40}{5}\right) \)
\( = 5\left(\frac{P – 40}{5}\right) + 40 \)
\( = (P – 40) + 40 \)
\( = P \).
2. Gosi na \((g \circ f)(T) = T \):
\( (g \circ f)(T) = g(f(T)) \)
Dochie \( f(T) = 5T + 40 \) n'ime \( g(P) \):
\( g(f(T)) = g(5T + 40) \)
\( = \frac{(5T + 40) – 40}{5} \)
\( = \frac{5T}{5} \)
\( = T \).
Ebe ọ bụ na \((f \circ g)(P) = P \) na \((g \circ f)(T) = T \), mgbe ahụ \(g \) bụ mgbanwe nke ọrụ \(f \).
Mmechi
Echiche nke nhazi ọrụ na ọrụ inverse dị oke mkpa na mgbakọ na mwepụ. Ha anaghị enyere anyị aka ịghọta mmekọrịta dị n'etiti ọrụ abụọ, kamakwa ha na-enye ntọala maka ojiji dị iche iche bara uru n'ụwa n'ezie, dị ka physics na injinia. Site n'ịmụ ihe atụ ndị dị n'elu, a na-atụ anya na ndị na-agụ akwụkwọ ga-enweta nghọta na itinye echiche abụọ a n'ọrụ nke ọma.
