Mehlala ea Lipotso Tse Buisanang ka Sebopeho sa Mesebetsi le Mesebetsi e Fetohileng
Lipalong tsa lipalo, mehopolo ea sebopeho sa mosebetsi le mesebetsi e fapaneng ke lihlooho tse peli tse amanang haufi-ufi tse bohlokoa bakeng sa kutloisiso e tsoetseng pele joalo ka calculus, tlhahlobo ea lipalo le khopolo-taba ea mosebetsi. Sengoloa sena se tla hlahloba likhopolo ka bobeli ka ho fana ka mehlala le lipuisano tse 'maloa tse bonolo ho li utloisisa. Sepheo ke ho thusa babali ho utloisisa hore na sebopeho sa mosebetsi le mekhoa e fapaneng ea mosebetsi li sebetsa joang ka tsela e sebetsang haholoanyane.
1. Sebopeho sa Mosebetsi
Sebopeho sa mosebetsi ke tshebetso ya ho kopanya mesebetsi e mmedi ho ba o mong. Haeba re na le mesebetsi e mmedi \( f(x) \) le \( g(x) \), jwale sebopeho sa mesebetsi ena ke \( (f \circ g)(x) \), e balwang “f sebopeho sa g ya x” kapa “f ya g ya x.” Sebopeho sena se hlaloswa e le ho sebedisa mosebetsi \( g(x) \) pele, ebe ho sebediswa mosebetsi \( f \) sephethong sa \( g(x) \).
Mohlala oa Potso ea 1:
Ho latela mesebetsi \( f(x) = 2x + 3 \) le \( g(x) = x^2 – 1 \). Fumana sebopeho sa \( (f \circ g)(x) \) le \( (g \circ f)(x) \).
Puisano:
1. Fumana \( (f \circ g)(x) \):
\( (f \circ g)(x) = f(g(x)) \)
\( = f(x^2 – 1) \)
Kenya sebaka sa \( x^2 – 1 \) ho \( f(x) \):
\( f(x^2 – 1) = 2(x^2 – 1) + 3 \)
\( = 2x^2 – 2 + 3 \)
\( = 2x^2 + 1 \)
Kahoo, \( (f \circ g)(x) = 2x^2 + 1 \).
2. Fumana \( (g \circ f)(x) \):
\( (g \potoloha f)(x) = g(f(x)) \)
\( = g(2x + 3) \)
Kenya sebaka sa \( 2x + 3 \) ho \( g(x) \):
\( g(2x + 3) = (2x + 3)^2 – 1 \)
Sebelisa boitsebiso ba quadratic ho bala \( (2x + 3)^2 \):
\( = 4x^2 + 12x + 9 – 1 \)
\( = 4x^2 + 12x + 8 \)
Kahoo, \( (g \circ f)(x) = 4x^2 + 12x + 8 \).
2. Mosebetsi o fapaneng
Mosebetsi o fapaneng ke mosebetsi o fetolang phello ea mosebetsi oa pele. Haeba \( f \) e le mosebetsi, joale phetoho ea \( f \), e ngotsoeng e le \( f^{-1} \), ke mosebetsi o khotsofatsang \( f(f^{-1}(x)) = x \) le \( f^{-1}(f(x)) = x \).
Ho fumana mosebetsi o fapaneng wa mosebetsi, re tlameha ho etsa tse latelang:
1. Nka sebaka sa \( f(x) \) ka \( y \).
2. Rarolla equation bakeng sa \( x \) ho latela \( y \).
3. Fetola diphetoho \( x \) le \( y \).
Mohlala oa Potso ea 2:
Ka lebaka la mosebetsi \( f(x) = 3x – 4 \), fumana hore o fapane, e leng \( f^{-1}(x) \).
Puisano:
1. Nka sebaka sa \( f(x) \) ka \( y \):
\( y = 3x – 4 \).
2. Rarolla \( x \) ho latela \( y \):
\( y = 3x – 4 \)
Kenya 4 mahlakoreng ka bobeli a equation:
\( y + 4 = 3x \)
Arola mahlakore ka bobeli a equation ka 3:
\( x = \frac{y + 4}{3} \)
3. Fetola diphetoho \( x \) le \( y \):
\( f^{-1}(x) = \frac{x + 4}{3} \)
Kahoo, phetoho ea \( f(x) = 3x – 4 \) ke \( f^{-1}(x) = \frac{x + 4}{3} \).
3. Mehlala ea Lipotso tse nang le Motsoako oa Sebopeho le Se fapaneng
Mohlala oa Potso ea 3:
Ha ho fanoe ka mesebetsi \( f(x) = x^3 + 2 \) le \( g(x) = \sqrt[3]{x – 2} \). Paka hore \( g(x) \) ke phetoho ya \( f(x) \).
Puisano:
Ho paka hore \( g(x) \) ke phetoho ya \( f(x) \), re tlameha ho bontsha hore \( (f \circ g)(x) = x \) le \( (g \circ f)(x) = x \).
1. Bontša hore \( (f \circ g)(x) = x \):
\( (f \circ g)(x) = f(g(x)) \)
Kenya sebaka sa \( g(x) = \sqrt[3]{x – 2} \) ho \( f(x) \):
\( f(g(x)) = f(\sqrt[3]{x – 2}) \)
\( = (\sqrt[3]{x – 2})^3 + 2 \)
Hobane \( (\sqrt[3]{x – 2})^3 = x – 2 \):
\( = (x – 2) + 2 \)
\( = x \).
2. Bontša hore \( (g \circ f)(x) = x \):
\( (g \potoloha f)(x) = g(f(x)) \)
Kenya sebaka sa \( f(x) = x^3 + 2 \) ho \( g(x) \):
\( g(f(x)) = g(x^3 + 2) \)
\( = \sqrt[3]{(x^3 + 2) – 2} \)
\( = \sqrt[3]{x^3} \)
\( = x \).
Kaha \( (f \circ g)(x) = x \) le \( (g \circ f)(x) = x \), joale \( g(x) \) ke phetoho ea \( f(x) \).
4. Litšebeliso Bophelong ba Letsatsi le Letsatsi
Mohlala oa Potso ea 4:
Rasaense o sebedisa mehlala e mmedi ya dipalo e hlalositsweng ke mesebetsi \( f(T) = 5T + 40 \) le \( g(P) = \frac{P – 40}{5} \), moo \( T \) e leng mocheso ka Celsius le \( P \) e leng kgatello ho Pascals. Fumana hore na mosebetsi \( g \) ke phetoho ya mosebetsi \( f \).
Puisano:
Ho paka hore \( g \) ke phetoho ea \( f \), re tlameha ho bontša hore \( (f \circ g)(P) = P \) le \( (g \circ f)(T) = T \).
1. Bontša hore \( (f \circ g)(P) = P \):
\( (f \circ g)(P) = f(g(P)) \)
Kenya sebaka sa \( g(P) = \frac{P – 40}{5} \) ho \( f(T) \):
\( f(g(P)) = f\left(\frac{P – 40}{5}\right) \)
\( = 5\left(\frac{P – 40}{5}\right) + 40 \)
\( = (P – 40) + 40 \)
\( = P \).
2. Bontša hore \( (g \circ f)(T) = T \):
\( (g \potoloha f)(T) = g(f(T)) \)
Kenya sebaka sa \( f(T) = 5T + 40 \) ho \( g(P) \):
\( g(f(T)) = g(5T + 40) \)
\( = \frac{(5T + 40) – 40}{5} \)
\( = \frac{5T}{5} \)
\( = T \).
Kaha \( (f \circ g)(P) = P \) le \( (g \circ f)(T) = T \), joale \( g \) ke phetoho ea ts'ebetso \( f \).
Qetello
Mehopolo ea sebopeho sa mosebetsi le mesebetsi e fapaneng e bohlokoa lipalo. Ha li re thuse feela ho utloisisa kamano pakeng tsa mesebetsi e 'meli, empa hape li fana ka motheo oa lits'ebetso tse fapaneng tse sebetsang lefatšeng la 'nete, joalo ka fisiks le boenjiniere. Ka ho ithuta mehlala e kaholimo, ho tšeptjoa hore babali ba tla utloisisa le ho sebelisa likhopolo tsena tse peli hamolemo.
