Question

If f(x) = XX+1 is invertible in its domain then fof-1(x) is equal to​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

$\sf{ \:The \: function \: f(x) = {x}^{2} + 1\: \: is \: invertible }$

TO DETERMINE

$\sf{(fo {f}^{ - 1} )(x) \: }$

CALCULATION

$\sf{ \: Let \: \: y = {f}^{ - 1} (x) \: }$

$\implies \sf{ f(y) = x}$

$\implies \sf{ {y}^{2} + 1 = x}$

$\implies \sf{ {y}^{2} = x - 1}$

$\implies \sf{ y = \sqrt{x - 1} }$

$\implies \sf{ {f}^{ - 1}(x) = \sqrt{x - 1} }$

Hence

$\sf{(fo {f}^{ - 1} )(x) \: }$

$= \sf{f \bigg( {f}^{ - 1} (x) \bigg)\: }$

$= \sf{f( \sqrt{x - 1} )\: }$

$= \sf{ { ( \sqrt{x - 1} )}^{2} + 1 \: }$

$= \sf{x - 1 + 1}$

$= \sf{x }$

RESULT

$\boxed{ \: \: \sf{(fo {f}^{ - 1} )(x) = x \: \: \: }}$

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LEARN MORE FROM BRAINLY

Let A={1,2,0,-1} B={1,3-1,-3} and

The function f = A to B f(x) =px+q

If f={ (1,1),(2,3),(0,-1),(-1,-3)}

find the value of p and q

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