Lipotso tsa Mehlala le Puisano ea Sebopeho sa Mosebetsi
Sebopeho sa mosebetsi ke mohopolo thutong ya dipalo moo mesebetsi e mmedi e kopantsweng ho ba o le mong. Haeba \( f \) le \( g \) e le mesebetsi e mmedi, jwale sebopeho sa \( f \) le \( g \) ke mosebetsi o motjha o hlaloswang e le \( (f \circ g)(x) \) e bolelang \( f(g(x)) \). Sehloohong sena, re tla buisana ka mathata a mmalwa a mehlala le mokhoa wa ho a rarolla a amanang le sebopeho sa mosebetsi.
1. Kutloisiso ea Motheo ea Sebopeho sa Ts'ebetso
Pele re kena lipotsong tsa mohlala, ha re utloisiseng ka bokhutšoanyane hore na sebopeho sa mosebetsi ke eng.
A re re ho na le mesebetsi e 'meli \( f \) le \( g \):
– Mosebetsi \( f \): \( x \mapsto f(x) \)
– Mosebetsi \( g \): \( x \mapsto g(x) \)
Sebopeho sa \( f \) le \( g \), se ngotsweng e le \( f \circ g \), ke mosebetsi o kgotsofatsang:
\[ (f \circ g)(x) = f(g(x)) \]
Mona, \( g(x) \) ke kenyelletso ya tshebetso \( f \).
2. Mohlala oa Potso ea 1
Potso:
Ha ho fanoe ka mosebetsi \( f(x) = 2x + 3 \) le mosebetsi \( g(x) = x – 5 \). Fumana \( (f \circ g)(x) \) le \( (g \circ f)(x) \).
Puisano:
A re baleng sebopeho sa pele \( (f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
Mohato oa pele, re kenya \( g(x) \) ho \( f(x) \):
\[ g(x) = x – 5 \]
\[ f(g(x)) = f(x – 5) \]
Mohato oa bobeli, re kenya \( x – 5 \) mosebetsing \( f \):
\[ f(x – 5) = 2(x – 5) + 3 \]
\[ = 2x – 10 + 3 \]
\[ = 2x – 7 \]
Kahoo, \( (f \circ g)(x) = 2x – 7 \).
Jwale ha re bale sebopeho sa bobedi \( (g \circ f)(x) \):
\[ (g \potoloha f)(x) = g(f(x)) \]
Mohato oa pele, re kenya \( f(x) \) ho \( g(x) \):
\[ f(x) = 2x + 3 \]
\[ g(f(x)) = g(2x + 3) \]
Mohato oa bobeli, re kenya \( 2x + 3 \) ts'ebetsong \( g \):
\[ g(2x + 3) = (2x + 3) – 5 \]
\[ = 2x + 3 – 5 \]
\[ = 2x – 2 \]
Kahoo, \( (g \circ f)(x) = 2x - 2 \).
3. Mohlala Potso ea 2: Sebopeho sa Mesebetsi e nang le Mesebetsi ea Quadratic
Potso:
Ha ho fanoe ka mosebetsi \( f(x) = x^2 + 1 \) le mosebetsi \( g(x) = 3x – 4 \). Fumana \( (f \circ g)(x) \) le \( (g \circ f)(x) \).
Puisano:
A re baleng sebopeho sa pele \( (f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
Mohato oa pele, re kenya \( g(x) \) ho \( f(x) \):
\[ g(x) = 3x – 4 \]
\[ f(g(x)) = f(3x – 4) \]
Mohato oa bobeli, re kenya \( 3x – 4 \) mosebetsing \( f \):
\[ f(3x – 4) = (3x – 4)^2 + 1 \]
\[ = (3x – 4)(3x – 4) + 1 \]
\[ = 9x^2 – 12x \cdot 2 + 16 + 1 \]
\[ = 9x^2 – 24x + 16 + 1 \]
\[ = 9x^2 – 24x + 17 \]
Kahoo, \( (f \circ g)(x) = 9x^2 – 24x + 17 \).
Jwale ha re bale sebopeho sa bobedi \( (g \circ f)(x) \):
\[ (g \potoloha f)(x) = g(f(x)) \]
Mohato oa pele, re kenya \( f(x) \) ho \( g(x) \):
\[ f(x) = x^2 + 1 \]
\[ g(f(x)) = g(x^2 + 1) \]
Mohato oa bobeli, re kenya \( x^2 + 1 \) mosebetsing \( g \):
\[ g(x^2 + 1) = 3(x^2 + 1) – 4 \]
\[ = 3x^2 + 3 – 4 \]
\[ = 3x^2 – 1 \]
Kahoo, \( (g \circ f)(x) = 3x^2 - 1 \).
4. Mohlala Potso ea 3: Sebopeho sa Mesebetsi ea Trigonometric
Potso:
Ho latela mosebetsi \( f(x) = \sin x \) le mosebetsi \( g(x) = x^2 \). Fumana \( (f \circ g)(x) \) le \( (g \circ f)(x) \).
Puisano:
A re baleng sebopeho sa pele \( (f \circ g)(x) \):
\[ (f \circ g)(x) = f(g(x)) \]
Mohato oa pele, re kenya \( g(x) \) ho \( f(x) \):
\[ g(x) = x^2 \]
\[ f(g(x)) = f(x^2) \]
Mohato oa bobeli, re kenya \( x^2 \) ts'ebetsong \( f \):
\[ f(x^2) = \sebe (x^2) \]
Kahoo, \( (f \circ g)(x) = \sin (x^2) \).
Jwale ha re bale sebopeho sa bobedi \( (g \circ f)(x) \):
\[ (g \potoloha f)(x) = g(f(x)) \]
Mohato oa pele, re kenya \( f(x) \) ho \( g(x) \):
\[ f(x) = \sebe x \]
\[ g(f(x)) = g(\sin x) \]
Mohato oa bobeli, re kenya \( \sin x \) ts'ebetsong \(g \):
\[ g(\sin x) = (\sin x)^2 \]
\[ = \sebe^2 x \]
Kahoo, \( (g \circ f)(x) = \sin^2 x \).
Qetello
Sebopeho sa mosebetsi ke mokhoa oa ho kopanya mesebetsi e 'meli hore e be mosebetsi o le mong. Ka mehlala e kaholimo, re ithutile hore ts'ebetso ea sebopeho sa mosebetsi e kenyelletsa ho nkela mosebetsi o mong sebaka ka o mong. Sephetho sa ho qetela sa sebopeho sa mosebetsi se itšetlehile haholo ka tatellano eo mesebetsi e sebelisoang ka eona pele.
Ho bohlokoa ho utloisisa hore \( (f \circ g)(x) \) ha se kamehla e tšoanang le \( (g \circ f)(x) \), 'me phapang ena e ka ba ea bohlokoa haholo lits'ebetsong tse fapaneng tsa lipalo le mahlale. Ka hona, ho utloisisa metheo le mokhoa oa ho bala sebopeho sa mesebetsi ho bohlokoa haholo ho mang kapa mang ea ithutang lipalo boemong bo mahareng kapa bo tsoetseng pele.
Re tšepa hore puisano le mehlala ea lipotso tse ka holimo li tla ba molemo 'me li thusa babali ho utloisisa ho hlophisoa ha mesebetsi.
