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Maths-

General

Easy

Question

If $f left parenthesis x right parenthesis equals fraction numerator e to the power of x plus e to the power of negative x end exponent over denominator 2 end fraction$ ,then the inverse function of $f left parenthesis x right parenthesis text is end text$

$\log_{e} (x + \sqrt{x^{2} + 1})$
$\log_{e} \sqrt{x^{2} + 1}$
$\log_{e} (x + \sqrt{x^{2} - 1})$
$\log_{e} (x - \sqrt{x^{2} - 1})$

hint Hint:

In this question, we have to find the inverse f(x) if $f left parenthesis x right parenthesis equals fraction numerator e to the power of x plus e to the power of negative x end exponent over denominator 2 end fraction$ . Firstly, we will consider f(x) to be y. Then we will solve the equation of getting the value of x from the equation. Further, as we get the value of x we can replace x with $f to the power of negative 1 left parenthesis x right parenthesis end exponent$ and on the other side, y with x.

The correct answer is: $log subscript e invisible function application open parentheses x plus square root of x squared minus 1 end root close parentheses$

$f left parenthesis x right parenthesis equals y equals fraction numerator e to the power of x plus e to the power of negative x end exponent over denominator 2 end fraction rightwards double arrow space 2 y space space equals e to the power of x plus e to the power of negative x end exponent rightwards double arrow space 2 y space space equals e to the power of x plus 1 over e to the power of x rightwards double arrow space 2 y space space equals fraction numerator e to the power of 2 x end exponent plus 1 over denominator e to the power of x end fraction L e t space e to the power of x space b e space t. space T h e n comma space rightwards double arrow space 2 y space space equals fraction numerator t squared plus 1 over denominator t end fraction rightwards double arrow 2 y t space space equals t squared plus 1 rightwards double arrow t squared minus 2 y t plus 1 equals space 0 rightwards double arrow t space equals fraction numerator 2 y plus-or-minus square root of 4 y squared minus 4 end root over denominator 2 end fraction space rightwards double arrow t equals y plus-or-minus square root of y squared minus 1 end root S u b s t i t u t i n g space b a c k space t space equals e to the power of x rightwards double arrow e to the power of x equals y plus-or-minus square root of y squared minus 1 end root P u t t i n n g space log space o n space b o t h space t h e space s i d e s comma space rightwards double arrow log space e to the power of x equals log space left parenthesis y plus-or-minus square root of y squared minus 1 end root right parenthesis rightwards double arrow x. space log space e space equals log space left parenthesis y plus-or-minus square root of y squared minus 1 end root right parenthesis rightwards double arrow x space equals fraction numerator log space left parenthesis y plus-or-minus square root of y squared minus 1 end root right parenthesis over denominator log space e end fraction rightwards double arrow x space equals log subscript e left parenthesis y plus-or-minus square root of y squared minus 1 end root right parenthesis C o n s i d e r i n g space p o s i t i v e comma space rightwards double arrow x space equals log subscript e left parenthesis y plus-or-minus square root of y squared minus 1 end root right parenthesis rightwards double arrow f to the power of negative 1 end exponent left parenthesis x right parenthesis equals log subscript e left parenthesis x plus-or-minus square root of x squared minus 1 end root right parenthesis$

Related Questions to study

If $f left parenthesis x right parenthesis equals x squared plus x comma x equals 10 comma delta x equals 0.1$ observe the following
$text l) end text delta f equals 2.11$
$text II) end text d f equals 2.1$
$text III) Relative error in end text x text is end text 1$
The true statements are

If $f left parenthesis x right parenthesis equals x squared plus x comma x equals 10 comma delta x equals 0.1$ observe the following
$text l) end text delta f equals 2.11$
$text II) end text d f equals 2.1$
$text III) Relative error in end text x text is end text 1$
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Which of the above statement/s is/are true.

Statement I : $f colon A not stretchy rightwards arrow B text is one - one and end text g colon B not stretchy rightwards arrow C text is a one-one function, then gof end text colon A not stretchy rightwards arrow C$ is one - one
Statement II : $text If end text f colon A not stretchy rightwards arrow B comma g colon B not stretchy rightwards arrow A$ are two functions such that $g o f equals I subscript A text and end text fog equals I subscript B comma text then end text f equals$ $g to the power of negative 1 end exponent$
Staatement III : $f left parenthesis x right parenthesis equals sec squared invisible function application x minus tan squared invisible function application x g left parenthesis x right parenthesis equals cosec squared invisible function application x minus cot squared invisible function application x comma text then end text f equals g$
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