Consider the function y=loga(x+x2+1 ),a>0,a=1. The inverse of the function

ay=(x+x2+1 )⇒a−y=x2+1−x⇒ay−a−y=2x⇒f−1(y)=x=2ay−a−y=2eylna −e−ylna =sin(ylnna ) (sinh(x)=2ex−e−x )

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Consider the function y= _{ a }{ ( x+ { { x }^{ 2 }+1 } ) ,a>

Question

Consider the function $y = lo g_{a} (x + x^{2} + 1), a > 0, a \neq = 1$ . The inverse of the function

does not exist

x = lo g_{a} (y + y^{2} + 1)

x = sin (y ln a)

x = cos h (- y ln \frac{1}{a})

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Levi Discussed

Consider the function

y = lo g_{a} (x + x^{2} + 1), a > 0, a \neq = 1

. The inverse of the function

6 mins ago

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6 mins ago

a^{y} = (x + x^{2} + 1) \Rightarrow a^{- y} = x^{2} + 1 - x \Rightarrow a^{y} - a^{- y} = 2 x

\Rightarrow f^{- 1} (y) = x = \frac{a ^{y} - a ^{- y}}{2} = \frac{e ^{y l n a} - e ^{- y l n a}}{2} = sin (y ln na) (sin h (x) = \frac{e ^{x} - e ^{- x}}{2})

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Question Text	Consider the function $y = lo g_{a} (x + x^{2} + 1), a > 0, a \neq = 1$ . The inverse of the function
Answer Type	Text solution:1
Upvotes	150