Fa'ata'ita'iga o Fesili ma Talanoaga o le Fa'atulagaga o Galuega
O le tuufaatasiga o galuega faatino o se manatu i le matematika lea e tuufaatasia ai galuega faatino e lua i le tasi. Afai o le \( f \) ma le \( g \) o galuega faatino e lua, o lona uiga o le tuufaatasiga o le \( f \) ma le \( g \) o se galuega faatino fou ua faauigaina o le \( (f \circ g)(x) \) o lona uiga o le \( f(g(x)) \). I totonu o lenei tusiga, o le a tatou talanoaina ai ni nai faataitaiga o faafitauli ma le auala e foia ai e fesoʻotaʻi ma le tuufaatasiga o galuega faatino.
1. Malamalamaaga Fa'avae o le Fa'atulagaga Fa'atino
A'o le'i o'o atu i fesili fa'ata'ita'i, se'i o tatou malamalama pu'upu'u po'o le a le fa'atulagaga o galuega (function composition).
Faapea e lua galuega faatino \( f \) ma le \( g \):
– Galuega \( f \) : \( x \mapsto f(x) \)
– Galuega \( g \) : \( x \mapsto g(x) \)
O le tuufaatasiga o le \( f \) ma le \( g \), tusia o le \( f \circ g \), o se galuega faatino e faamalieina ai:
\[ (f \circ g)(x) = f(g(x)) \]
O iinei, o le \( g(x) \) o le input i le function \( f \).
2. Fa'ata'ita'iga Fesili 1
Fesili:
I le tuuina atu o le galuega faatino \( f(x) = 2x + 3 \) ma le galuega faatino \( g(x) = x – 5 \). Fuafua \( (f \circ g)(x) \) ma le \( (g \circ f)(x) \).
Talanoaga:
Se'i o tatou fuafuaina le tuufaatasiga muamua \( (f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
O le laasaga muamua, tatou te fa'aofi le \( g(x) \) i totonu o le \( f(x) \):
\[ g(x) = x – 5 \]
\[ f(g(x)) = f(x – 5) \]
Laasaga lona lua, tatou te fa'aofiina le \( x – 5 \) i totonu o le galuega faatino \( f \):
\[ f(x – 5) = 2(x – 5) + 3 \]
\[ = 2x – 10 + 3 \]
\[ = 2x – 7 \]
O lea la, \( (f \circ g)(x) = 2x – 7 \).
Ia, se’i o tatou fuafuaina le tuufaatasiga lona lua \( (g \circ f)(x) \):
\[ (g \circ f)(x) = g(f(x)) \]
Laasaga muamua, tatou te fa'aofi le \( f(x) \) i totonu o le \( g(x) \):
\[ f(x) = 2x + 3 \]
\[ g(f(x)) = g(2x + 3) \]
Laasaga lona lua, tatou te fa'aofiina le \( 2x + 3 \) i totonu o le galuega faatino \( g \):
\[ g(2x + 3) = (2x + 3) – 5 \]
\[ = 2x + 3 – 5 \]
\[ = 2x – 2 \]
O lea la, \( (g \circ f)(x) = 2x – 2 \).
3. Fa'ata'ita'iga Fesili 2: Fa'atulagaina o Galuega Fa'atino ma Galuega Fa'atino Fa'atafafā
Fesili:
I le tuuina atu o le galuega faatino \( f(x) = x^2 + 1 \) ma le galuega faatino \( g(x) = 3x – 4 \). Fuafua \( (f \circ g)(x) \) ma le \( (g \circ f)(x) \).
Talanoaga:
Se'i o tatou fuafuaina le tuufaatasiga muamua \( (f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
O le laasaga muamua, tatou te fa'aofi le \( g(x) \) i totonu o le \( f(x) \):
\[ g(x) = 3x – 4 \]
\[ f(g(x)) = f(3x – 4) \]
Laasaga lona lua, tatou te fa'aofiina le \( 3x – 4 \) i totonu o le galuega faatino \( f \):
\[ f(3x – 4) = (3x – 4)^2 + 1 \]
\[ = (3x – 4)(3x – 4) + 1 \]
\[ = 9x^2 – 12x \cdot 2 + 16 + 1 \]
\[ = 9x^2 – 24x + 16 + 1 \]
\[ = 9x^2 – 24x + 17 \]
O lea la, \( (f \circ g)(x) = 9x^2 – 24x + 17 \).
Ia, se’i o tatou fuafuaina le tuufaatasiga lona lua \( (g \circ f)(x) \):
\[ (g \circ f)(x) = g(f(x)) \]
Laasaga muamua, tatou te fa'aofi le \( f(x) \) i totonu o le \( g(x) \):
\[ f(x) = x^2 + 1 \]
\[ g(f(x)) = g(x^2 + 1) \]
Laasaga lona lua, tatou te fa'aofiina le \( x^2 + 1 \) i totonu o le galuega faatino \( g \):
\[ g(x^2 + 1) = 3(x^2 + 1) – 4 \]
\[ = 3x^2 + 3 – 4 \]
\[ = 3x^2 – 1 \]
O lea la, \( (g \circ f)(x) = 3x^2 – 1 \).
4. Fa'ata'ita'iga Fesili 3: Fa'atulagaga o Galuega Fa'atino Trigonometric
Fesili:
I le tuuina atu o le galuega faatino \( f(x) = \sin x \) ma le galuega faatino \( g(x) = x^2 \). Fuafua \( (f \circ g)(x) \) ma le \( (g \circ f)(x) \).
Talanoaga:
Se'i o tatou fuafuaina le tuufaatasiga muamua \( (f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
O le laasaga muamua, tatou te fa'aofi le \( g(x) \) i totonu o le \( f(x) \):
\[ g(x) = x^2 \]
\[ f(g(x)) = f(x^2) \]
Laasaga lona lua, tatou te fa'aofiina le \( x^2 \) i totonu o le galuega faatino \( f \):
\[ f(x^2) = \sin (x^2) \]
O lea la, \( (f \circ g)(x) = \sin (x^2) \).
Ia, se’i o tatou fuafuaina le tuufaatasiga lona lua \( (g \circ f)(x) \):
\[ (g \circ f)(x) = g(f(x)) \]
Laasaga muamua, tatou te fa'aofi le \( f(x) \) i totonu o le \( g(x) \):
\[ f(x) = \sin x \]
\[ g(f(x)) = g(\sin x) \]
Laasaga lona lua, tatou te fa'aofiina le \( \sin x \) i totonu o le galuega faatino \( g \):
\[ g(\sin x) = (\sin x)^2 \]
\[ = \sin^2 x \]
O lea la, \( (g \circ f)(x) = \sin^2 x \).
I'uga
O le tuufaatasiga o galuega faatino o se auala e tuufaatasia ai galuega faatino e lua i se galuega faatino e tasi. E ala i faataitaiga o loʻo i luga, na tatou iloa ai o le faagasologa o le tuufaatasiga o galuega faatino e aofia ai le suiina o le tasi galuega faatino i le isi. O le taunuuga mulimuli o le tuufaatasiga o galuega faatino e faalagolago tele i le faasologa o le faaaogaina muamua o galuega faatino.
E taua le malamalama e lē tutusa i taimi uma le \( (f \circ g)(x) \) ma le \( (g \circ f)(x) \), ma e mafai ona matuā tāua tele lenei eseesega i le tele o faʻaoga o le matematika ma le saienisi. O le mea lea, o le malamalama i mea faavae ma le auala e fuafua ai le tuufaatasiga o galuega faatino e matuā tāua tele mo soʻo se tasi e suʻesuʻeina le matematika i se tulaga ogatotonu poʻo se tulaga maualuga.
Faʻamoemoe o le a aoga le talanoaga ma faʻataʻitaʻiga o fesili o loʻo i luga ma fesoasoani i le au faitau e malamalama i le faʻatulagaina o galuega tauave.
