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The inverse of the function f(x)=(e^x-e^...

The inverse of the function `f(x)=(e^x-e^(-x))/(e^x+e^(-x))+2` is given by

Text Solution

Verified by Experts

The correct Answer is:
`f^(-1)(x)=log_(e)((x-1)/(3-x))^(1//2)`
`y=(e^(x)-e^(-x))/(e^(x)+e^(-x))+2=(e^(2x)-1)/(e^(2x)+1)+2`
or `e^(2x)=(1-y)/(y-3)=(y-1)/(3-y)`
or `x=(1)/(2)log_(e)((y-1)/(3-y))`
or `f^(-1)(y)=log_(e)((y-1)/(3-y))^(1//2)`
or `f^(-1)(x)=log_(e)((x-1)/(3-x))^(1//2)`
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Knowledge Check

  • The inverse of the function f(x)=(e^(x)-2e^(-x))/(e^(x)+2e^(-x))+1 is

    A
    `log_(10)((2x)/(2-x))`
    B
    `log_(10)((x)/(2-x))`
    C
    `log_(e )((2x)/(2-x))^(1//2)`
    D
    `In((2x)/(2-x))^(1//2)`

  • The inverse of the function y=(e^(2x)-e^(-2x))/(e^(2x)+e^(-2x)) is/an

    A
    `log_(e )((1+2x)/(1-2x))`
    B
    `1/4 log_(e )((1-x)/(1+x))`
    C
    `1/4 log_(e )((1+x)/(1-x))`
    D
    an odd function

  • The inverse of the function log_e x is

    A
    `10^x`
    B
    `10^(-x)`
    C
    `e^x`
    D
    `e^(-x)`

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