Solution to Show that if π‘Žπ‘‘βˆ’π‘π‘ β‰  0, then the function 𝑓(π‘₯)=π‘Žπ‘₯+𝑏/𝑐π‘₯+𝑑 is one-to-one and find its … - Sikademy
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Show that if π‘Žπ‘‘βˆ’π‘π‘ β‰  0, then the function 𝑓(π‘₯)=π‘Žπ‘₯+𝑏/𝑐π‘₯+𝑑 is one-to-one and find its inverse.

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cx+d\not=0=>x\not=-d/c

Domain:Β (-\infin, -d/c)\cup(-d/c, \infin)

ReplaceΒ f(x)Β withΒ y


y=\dfrac{ax+b}{cx+d}

SwitchΒ xΒ andΒ y


x=\dfrac{ay+b}{cy+d}

Solve forΒ y


cxy+dx=ay+b

y=\dfrac{b-dx}{cx-a}

cx-a\not=0=>x\not=c/a


f^{-1}\circ f=\dfrac{b-d(\dfrac{ax+b}{cx+d})}{c(\dfrac{ax+b}{cx+d})-a}

=\dfrac{bcx+bd-adx-bd}{acx+bc-acx-ad}

=\dfrac{bcx-adx}{bc-ad}=x, bc\not=ad=>ad-bc\not=0

Therefore Β ifΒ ad-bc\not=0,Β then the functionΒ f(x)=\dfrac{ax+b}{cx+d}Β is one-to-one and find its inverse.

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Question ID: mtid-5-stid-8-sqid-1342-qpid-1080