Mibvunzo Yemuenzaniso Inotsanangura Kuumbwa Kwemabasa uye Mabasa Akasiyana
Mumasvomhu, pfungwa dzekushanda kwemabasa uye mabasa ekuchinja-chinja inyaya mbiri dzakabatana dzakakosha pakunzwisisa kwepamusoro senge calculus, mathematical analysis, uye function theory. Chinyorwa chino chichaongorora pfungwa dzese nekupa mienzaniso yakati wandei iri nyore kunzwisisa uye hurukuro. Chinangwa ndechekubatsira vaverengi kunzwisisa kuti kushanda kwemabasa uye mabasa ekuchinja-chinja anoshanda sei nenzira inoshanda.
1. Kuumbwa kwebasa
Kuumbwa kwebasa ndiko kushanda kwekubatanidza mabasa maviri kuita rimwe. Kana tiine mabasa maviri \( f(x) \) uye \( g(x) \), saka kuumbwa kwebasa iri ndi \( (f \circ g)(x) \), izvo zvinoverengwa "f kuumbwa g ya x" kana "f ya g ya x." Kuumbwa uku kunotsanangurwa sekushandisa basa \( g(x) \) kutanga, wozoshandisa basa \( f \) kumhedzisiro ya \( g(x) \).
Muenzaniso Mubvunzo 1:
Zvichienderana nemabasa \( f(x) = 2x + 3 \) uye \( g(x) = x^2 – 1 \). Tsvaga maumbirwo e \( (f \circ g)(x) \) uye \( (g \circ f)(x) \).
Kukurukurirana:
1. Sarudza \( (f \circ g)(x) \):
\( (f \circ g)(x) = f(g(x)) \)
\( = f(x^2 – 1) \)
Isa \( x^2 – 1 \) mu \( f(x) \):
\( f(x^2 – 1) = 2(x^2 – 1) + 3 \)
\( = 2x^2 – 2 + 3 \)
\( = 2x^2 + 1 \)
Saka, \( (f \circ g)(x) = 2x^2 + 1 \).
2. Sarudza \( (g \circ f)(x) \):
\( (g \denderedzwa f)(x) = g(f(x)) \)
\( = g(2x + 3) \)
Isa \( 2x + 3 \) mu \( g(x) \):
\( g(2x + 3) = (2x + 3)^2 – 1 \)
Shandisa quadratic identity kuverenga \( (2x + 3)^2 \):
\( = 4x^2 + 12x + 9 – 1 \)
\( = 4x^2 + 12x + 8 \)
Saka, \( (g \circ f)(x) = 4x^2 + 12x + 8 \).
2. Basa reInverse
Basa rinopindurwa ibasa rinopindurwa rinoshandura simba rebasa rekutanga. Kana \( f \) riri basa, saka rekuti \( f \), rakanyorwa se \( f^{-1} \), ibasa rinogutsa \( f(f^{-1}(x)) = x \) uye \( f^{-1}(f(x)) = x \).
Kuti tiwane basa re "inverse" rebasa, tinofanira kuita zvinotevera:
1. Tsiva \( f(x) \) na \( y \).
2. Gadzirisa equation ye \( x \) maererano ne \( y \).
3. Chinjana mavariables \( x \) uye \( y \).
Muenzaniso Mubvunzo 2:
Zvichienderana nebasa \( f(x) = 3x – 4 \), tsvaga zvinopesana naro, kureva \( f^{-1}(x) \).
Kukurukurirana:
1. Tsiva \( f(x) \) na \( y \):
\( y = 3x – 4 \).
2. Gadzirisa \( x \) maererano ne \( y \):
\( y = 3x – 4 \)
Wedzera 4 kumativi ese e equation:
\( y + 4 = 3x \)
Govanisa mativi ese ari maviri e equation ne3:
\( x = \frac{y + 4}{3} \)
3. Chinjana mavariables \( x \) uye \( y \):
\( f^{-1}(x) = \frac{x + 4}{3} \)
Saka, zvinopesana ne \( f(x) = 3x – 4 \) ndi \( f^{-1}(x) = \frac{x + 4}{3} \).
3. Mibvunzo yemuenzaniso ine musanganiswa weKuumbwa neInverse
Muenzaniso Mubvunzo 3:
Zvichienderana nemabasa \( f(x) = x^3 + 2 \) uye \( g(x) = \sqrt[3]{x – 2} \). Ratidza kuti \( g(x) \) inopesana ne \( f(x) \).
Kukurukurirana:
Kuti tiratidze kuti \( g(x) \) inopesana na \( f(x) \), tinofanira kuratidza kuti \( (f \circ g)(x) = x \) uye \( (g \circ f)(x) = x \).
1. Ratidza kuti \( (f \circ g)(x) = x \):
\( (f \circ g)(x) = f(g(x)) \)
Isa \( g(x) = \sqrt[3]{x – 2} \) mu \( f(x) \):
\( f(g(x)) = f(\sqrt[3]{x – 2}) \)
\( = (\sqrt[3]{x – 2})^3 + 2 \)
Nekuti \( (\sqrt[3]{x – 2})^3 = x – 2 \):
\( = (x – 2) + 2 \)
\( = x \).
2. Ratidza kuti \( (g \circ f)(x) = x \):
\( (g \denderedzwa f)(x) = g(f(x)) \)
Isa \( f(x) = x^3 + 2 \) mu \( g(x) \):
\( g(f(x)) = g(x^3 + 2) \)
\( = \sqrt[3]{(x^3 + 2) – 2} \)
\( = \sqrt[3]{x^3} \)
\( = x \).
Sezvo \( (f \circ g)(x) = x \) uye \( (g \circ f)(x) = x \), saka \( g(x) \) ndiyo inverse ye \( f(x) \).
4. Mashandisirwo muHupenyu hweZuva Nezuva
Muenzaniso Mubvunzo 4:
Nyanzvi yesainzi inoshandisa mamodheru maviri emasvomhu anotsanangurwa nemabasa \( f(T) = 5T + 40 \) uye \( g(P) = \frac{P – 40}{5} \), apo \( T \) iri tembiricha muCelsius uye \( P \) iri kumanikidzwa muPascals. Sarudza kana basa \( g \) riri inverse yebasa \( f \).
Kukurukurirana:
Kuti tiratidze kuti \( g \) inopesana na \( f \), tinofanira kuratidza kuti \( (f \circ g)(P) = P \) uye \( (g \circ f)(T) = T \).
1. Ratidza kuti \( (f \circ g)(P) = P \):
\( (f \circ g)(P) = f(g(P)) \)
Isa \( g(P) = \frac{P – 40}{5} \) mu \( f(T) \):
\( f(g(P)) = f\left(\frac{P – 40}{5}\right) \)
\( = 5\kuruboshwe(\frac{P – 40}{5}\kurudyi) + 40 \)
\( = (P – 40) + 40 \)
\( = P \).
2. Ratidza kuti \( (g \circ f)(T) = T \):
\( (g \denderedzwa f)(T) = g(f(T)) \)
Isa \( f(T) = 5T + 40 \) mu \( g(P) \):
\( g(f(T)) = g(5T + 40) \)
\( = \frac{(5T + 40) – 40}{5} \)
\( = \frac{5T}{5} \)
\( = T \).
Sezvo \( (f \circ g)(P) = P \) uye \( (g \circ f)(T) = T \), saka \( g \) ndiyo inverse yebasa \( f \).
Mhedziso
Pfungwa dzekuumbwa kwebasa uye mabasa akasiyana-siyana dzakakosha mumasvomhu. Hazvingobatsiri chete kunzwisisa hukama huripo pakati pemabasa maviri, asiwo zvinopa hwaro hwemashandisirwo akasiyana-siyana anoshanda munyika chaiyo, senge fizikisi neinjiniya. Nekudzidza mienzaniso iri pamusoro, zvinotarisirwa kuti vaverengi vachanzwisisa zviri nani uye vachashandisa pfungwa idzi mbiri.
