Mibvunzo yeMienzaniso neKukurukurirana kweKuumbwa kweBasa
Kuumbwa kwebasa ipfungwa mumasvomhu umo mabasa maviri anobatanidzwa kuita rimwe. Kana \( f \) uye \( g \) ari mabasa maviri, saka kuumbwa kwe \( f \) uye \( g \) ibasa idzva rinotsanangurwa se \( (f \circ g)(x) \) zvinoreva \( f(g(x)) \). Muchinyorwa chino, tichakurukura mienzaniso yakati wandei yematambudziko uye maitiro ekugadzirisa ane chekuita nekuumbwa kwebasa.
1. Kunzwisisa Kwekutanga Kwekuumbwa Kwebasa
Tisati tapinda mumibvunzo yemuenzaniso, ngatinzwisisei muchidimbu kuti chii chinonzi function composition.
Ngatitii pane mabasa maviri \( f \) uye \( g \):
– Basa \( f \): \( x \mapsto f(x) \)
– Basa \( g \): \( x \mapsto g(x) \)
Kuumbwa kwe \( f \) na \( g \), kwakanyorwa se \( f \circ g \), ibasa rinogutsa:
\[ (f \circ g)(x) = f(g(x)) \]
Pano, \( g(x) \) ndiyo inopinzwa basa \( f \).
2. Muenzaniso Mubvunzo 1
Mubvunzo:
Zvichienderana nebasa \( f(x) = 2x + 3 \) uye basa \( g(x) = x – 5 \). Sarudza \( (f \circ g)(x) \) uye \( (g \circ f)(x) \).
Kukurukurirana:
Ngativerengei musanganiswa wekutanga \( (f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
Danho rekutanga, tinoisa \( g(x) \) mu \( f(x) \):
\[ g(x) = x – 5 \]
\[ f(g(x)) = f(x – 5) \]
Danho rechipiri, tinopinda \( x – 5 \) mubasa \( f \):
\[ f(x – 5) = 2(x – 5) + 3 \]
\[ = 2x – 10 + 3 \]
\[ = 2x – 7 \]
Saka, \( (f \circ g)(x) = 2x – 7 \).
Zvino ngativerengei chimiro chechipiri \( (g \circ f)(x) \):
\[ (g \denderedzwa f)(x) = g(f(x)) \]
Danho rekutanga, tinoisa \( f(x) \) mu \( g(x) \):
\[ f(x) = 2x + 3 \]
\[ g(f(x)) = g(2x + 3) \]
Danho rechipiri, tinoisa \( 2x + 3 \) mubasa \( g \):
\[ g(2x + 3) = (2x + 3) – 5 \]
\[ = 2x + 3 – 5 \]
\[ = 2x – 2 \]
Saka, \( (g \circ f)(x) = 2x - 2 \).
3. Muenzaniso Mubvunzo 2: Kuumbwa kweMabasa ane Mabasa eQuadratic
Mubvunzo:
Zvichienderana nebasa \( f(x) = x^2 + 1 \) uye basa \( g(x) = 3x – 4 \). Sarudza \( (f \circ g)(x) \) uye \( (g \circ f)(x) \).
Kukurukurirana:
Ngativerengei musanganiswa wekutanga \( (f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
Danho rekutanga, tinoisa \( g(x) \) mu \( f(x) \):
\[ g(x) = 3x – 4 \]
\[ f(g(x)) = f(3x – 4) \]
Danho rechipiri, tinopinda \( 3x – 4 \) mubasa \( f \):
\[ f(3x – 4) = (3x – 4)^2 + 1 \]
\[ = (3x – 4)(3x – 4) + 1 \]
\[ = 9x^2 – 12x \cdot 2 + 16 + 1 \]
\[ = 9x^2 – 24x + 16 + 1 \]
\[ = 9x^2 – 24x + 17 \]
Saka, \( (f \circ g)(x) = 9x^2 – 24x + 17 \).
Zvino ngativerengei chimiro chechipiri \( (g \circ f)(x) \):
\[ (g \denderedzwa f)(x) = g(f(x)) \]
Danho rekutanga, tinoisa \( f(x) \) mu \( g(x) \):
\[ f(x) = x^2 + 1 \]
\[ g(f(x)) = g(x^2 + 1) \]
Danho rechipiri, tinopinda \( x^2 + 1 \) mubasa \( g \):
\[ g(x^2 + 1) = 3(x^2 + 1) – 4 \]
\[ = 3x^2 + 3 – 4 \]
\[ = 3x^2 – 1 \]
Saka, \( (g \circ f)(x) = 3x^2 - 1 \).
4. Muenzaniso Mubvunzo 3: Kuumbwa kweMabasa eTrigonometric
Mubvunzo:
Zvichienderana nebasa \( f(x) = \sin x \) uye basa \( g(x) = x^2 \). Sarudza \( (f \circ g)(x) \) uye \( (g \circ f)(x) \).
Kukurukurirana:
Ngativerengei musanganiswa wekutanga \( (f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
Danho rekutanga, tinoisa \( g(x) \) mu \( f(x) \):
\[ g(x) = x^2 \]
\[ f(g(x)) = f(x^2) \]
Danho rechipiri, tinopinda \( x^2 \) mubasa \( f \):
\[ f(x^2) = \chivi (x^2) \]
Saka, \( (f \circ g)(x) = \sin (x^2) \).
Zvino ngativerengei chimiro chechipiri \( (g \circ f)(x) \):
\[ (g \denderedzwa f)(x) = g(f(x)) \]
Danho rekutanga, tinoisa \( f(x) \) mu \( g(x) \):
\[ f(x) = \chivi x \]
\[ g(f(x)) = g(\chivi x) \]
Danho rechipiri, tinoisa \( \sin x \) mubasa \(g \):
\[ g(\chivi x) = (\chivi x)^2 \]
\[ = \chivi^2 x \]
Saka, \( (g \circ f)(x) = \sin^2 x \).
Mhedziso
Kuumbwa kwebasa inzira yekubatanidza mabasa maviri kuita basa rimwe chete. Kuburikidza nemienzaniso iri pamusoro, takadzidza kuti maitiro ekuumbwa kwebasa anosanganisira kutsiva basa rimwe nerimwe. Mhedzisiro yekupedzisira yekuumbwa kwebasa inoenderana zvakanyanya nekurongeka kwekushandiswa kwebasa kutanga.
Zvakakosha kunzwisisa kuti \( (f \circ g)(x) \) haisi nguva dzose yakafanana ne \( (g \circ f)(x) \), uye musiyano uyu unogona kuva wakakosha zvikuru mumashandisirwo akasiyana-siyana emasvomhu nesainzi. Nokudaro, kunzwisisa zvinhu zvekutanga uye maverengerwo emabasa kwakakosha zvikuru kune chero munhu ari kudzidza masvomhu padanho repakati kana repamusoro.
Tinovimba kuti hurukuro nemienzaniso yemibvunzo iri pamusoro apa inobatsira uye inobatsira vaverengi kunzwisisa kuumbwa kwemashandiro.
