Su'aalo Tusaale ah oo Ka Hadlaya Qaab-dhismeedka Hawlaha iyo Hawlaha Rogmada ah
Xisaabta, fikradaha halabuurka shaqada iyo hawlaha rogan waa laba mowduuc oo si dhow isugu xiran oo muhiim u ah fahamka horumarsan sida xisaabinta, falanqaynta xisaabta, iyo aragtida shaqada. Maqaalkani wuxuu sahamin doonaa labada fikradood isagoo bixinaya tusaalooyin iyo doodo dhowr ah oo si fudud loo fahmi karo. Hadafku waa in akhristayaasha laga caawiyo inay fahmaan sida halabuurka shaqada iyo kuwa rogan ee shaqeynaya ay u shaqeeyaan si wax ku ool ah.
1. Halabuurka Shaqada
Halabuurka shaqo waa hawlgalka isku darka laba shaqo oo mid ah. Haddii aan leenahay laba shaqo oo \( f(x) \) iyo \( g(x) \), markaa halabuurka shaqooyinkan waa \( (f \circ g)(x) \), kaas oo loo akhriyo "f halabuurka g ee x" ama "f ee g ee x." Halabuurkan waxaa lagu qeexaa inuu marka hore ku dabaqayo shaqada \( g(x) \) ka dibna uu ku dabaqayo shaqada \( f \) natiijada \( g(x) \).
Su'aal Tusaale 1aad:
Marka la eego shaqooyinka \( f(x) = 2x + 3 \) iyo \( g(x) = x^2 – 1 \). Soo hel halabuurka \( (f \circ g)(x) \) iyo \( (g \circ f)(x) \).
Dood:
1. Go'aami \( (f \circ g)(x) \):
\( (f \circle g)(x) = f(g(x)) \)
\( = f(x^2 – 1) \)
Ku beddel \( x^2 – 1 \) una beddel \( f(x) \):
\( f(x^2 – 1) = 2(x^2 – 1) + 3 \)
\( = 2x^2 – 2 + 3 \)
\( = 2x^2 + 1 \)
Markaa, \( (f \circ g)(x) = 2x^2 + 1 \).
2. Go'aami \( (g \circ f)(x) \):
\( (g \circle f)(x) = g(f(x)) \)
\( = g(2x + 3) \)
Ku beddel \( 2x + 3 \) una beddel \( g(x) \):
\( g(2x + 3) = (2x + 3)^2 – 1 \)
Adeegso aqoonsiga labajibbaaran si aad u xisaabiso \( (2x + 3)^2 \):
\( = 4x^2 + 12x + 9 – 1 \)
\( = 4x^2 + 12x + 8 \)
Markaa, \((g \circ f)(x) = 4x^2 + 12x + 8 \).
2. Shaqada rogan
Shaqada rogan waa shaqo rogan oo beddesha saameynta shaqada asalka ah. Haddii \( f \) uu yahay shaqo, markaas roganka \( f \), oo loo qoray \( f^{-1} \), waa shaqo qancisa \( f(f^{-1}(x)) = x \) iyo \( f^{-1}(f(x)) = x \).
Si loo helo shaqada rogan ee shaqada, waa inaan sameynaa waxyaabaha soo socda:
1. Ku beddel \( f(x) \) \( y \).
2. Xalli isle'egta \( x \) marka loo eego \( y \).
3. Isweydaarso doorsoomayaasha \( x \) iyo \( y \).
Su'aal Tusaale 2aad:
Marka la eego shaqada \( f(x) = 3x – 4 \), hel rogaalkeeda, kaas oo ah \( f^{-1}(x) \).
Dood:
1. Ku beddel \( f(x) \) \( y \):
\( y = 3x – 4 \).
2. Xal u hel \( x \) marka loo eego \( y \):
\( y = 3x – 4 \)
Ku dar 4 labada dhinac ee isla'egta:
\( y + 4 = 3x \)
Labada dhinac ee isla'egta u qaybi 3:
\( x = \frac{y + 4}{3} \)
3. Isweydaarso doorsoomayaasha \( x \) iyo \( y \):
\( f^{-1}(x) = \frac{x + 4}{3} \)
Markaa, lidkeeda \( f(x) = 3x – 4 \) waa \( f^{-1}(x) = \frac{x + 4}{3} \).
3. Su'aalo Tusaale ah oo leh Isku-darka Halabuurka iyo Rogaal-celinta
Su'aal Tusaale 3aad:
Marka la eego shaqooyinka \( f(x) = x^3 + 2 \) iyo \( g(x) = \sqrt[3]{x – 2} \). Caddee in \( g(x) \) uu yahay lidkeeda \( f(x) \).
Dood:
Si loo caddeeyo in \( g(x) \) uu yahay lidkiisa \( f(x) \), waa inaan tusnaa in \( (f \circ g)(x) = x \) iyo \( (g \circ f)(x) = x \).
1. Muuji in \( (f \circ g)(x) = x \):
\( (f \circle g)(x) = f(g(x)) \)
Ku beddel \( g(x) = \sqrt[3]{x – 2} \) una beddel \( f(x) \):
\( f(g(x)) = f(\sqrt[3]{x – 2}) \)
\( = (\sqrt[3]{x – 2})^3 + 2 \)
Sababtoo ah \( (\sqrt[3]{x – 2})^3 = x – 2 \):
\( = (x – 2) + 2 \)
\( = x \).
2. Muuji in \( (g \circ f)(x) = x \):
\( (g \circle f)(x) = g(f(x)) \)
Ku beddel \( f(x) = x^3 + 2 \) una beddel \( g(x) \):
\( g(f(x)) = g(x^3 + 2) \)
\( = \sqrt[3]{(x^3 + 2) – 2} \)
\( = \sqrt[3]{x^3} \)
\( = x \).
Maadaama \((f \circ g)(x) = x \) iyo \((g \circ f)(x) = x \), markaas \(g(x) \) waa lidkeeda \( f(x) \).
4. Codsiyada Nolosha Maalinlaha ah
Su'aal Tusaale 4aad:
Saynisyahanku wuxuu adeegsadaa laba qaab oo xisaabeed oo lagu sharraxay hawlaha \( f(T) = 5T + 40 \) iyo \( g(P) = \frac{P – 40}{5} \), halkaas oo \( T \) uu yahay heerkulka Celsius iyo \( P \) uu yahay cadaadiska Pascals. Go'aami in shaqada \( g \) ay tahay beddelka shaqada \( f \).
Dood:
Si aan u caddeyno in \( g \) uu yahay lidkiisa \( f \), waa inaan tusnaa in \( (f \circ g)(P) = P \) iyo \( (g \circ f)(T) = T \).
1. Muuji in \( (f \circ g)(P) = P \):
\( (f \circle g)(P) = f(g(P)) \)
Ku beddel \( g(P) = \frac{P – 40}{5} \) una beddel \( f(T) \):
\( f(g(P)) = f\left(\frac{P – 40}{5}\right) \)
\( = 5\left(\frac{P – 40}{5}\right) + 40 \)
\( = (P – 40) + 40 \)
\( = P \).
2. Muuji in \( (g \circ f)(T) = T \):
\( (g \circle f)(T) = g(f(T)) \)
Ku beddel \( f(T) = 5T + 40 \) oo u beddel \( g(P) \):
\( g(f(T)) = g(5T + 40) \)
\( = \frac{(5T + 40) – 40}{5} \)
\( = \frac{5T}{5} \)
\( = T \).
Maadaama \((f \circ g)(P) = P \) iyo \((g \circ f)(T) = T \), markaas \(g \) waa lidkeeda shaqada \( f \).
Gabagabo
Fikradaha halabuurka shaqada iyo shaqooyinka rogan ayaa muhiim u ah xisaabta. Kaliya ma caawiyaan inaan fahanno xiriirka ka dhexeeya laba shaqo, laakiin sidoo kale waxay bixiyaan aasaaska codsiyada kala duwan ee wax ku oolka ah ee adduunka dhabta ah, sida fiisigiska iyo injineernimada. Marka la barto tusaalooyinka kor ku xusan, waxaa la rajeynayaa in akhristayaashu ay heli doonaan faham wanaagsan iyo ku dabaqid labada fikradood.
