Show that the function f (x) = x3 andg (x) = x^1/3 for all x e R are i – InterviewSolution

1.

Show that the function f (x) = x3 andg (x) = x^1/3 for all x e R are inverse of one another.

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SOLUTION

TO PROVE

$\displaystyle \sf{Two \: functions \: \: f(x) = {x}^{3} \: \: and \: \: <klux>G</klux>(x) = {x}^{ \frac{1}{3} } \: for \: all \: x \in \mathbb{ R} }$

are inverse of one another

CONCEPT TO BE IMPLEMENTED

Two GIVEN real valued function f(x) & g(x) are said to be inverse of one another if

$\sf{(f \circ g)(x) = (g \circ f)(x) = x}$

EVALUATION

Here the given two functions are

$\displaystyle \sf{ f(x) = {x}^{3} \: \: and \: \: g(x) = {x}^{ \frac{1}{3} } \: for \: all \: x \in \mathbb{ R} }$

Now

$\sf{(f \circ g)(x)}$

$\sf{ = f(g(x))}$

$\displaystyle \sf{ = \: f \big( {x}^{ \frac{1}{3} } \big) }$

$\displaystyle \sf{ = {\big( {x}^{ \frac{1}{3} } \big)}^{3} }$

$= \sf{x}$

Again

$\displaystyle \sf{ (g \circ f)(x) }$

$\displaystyle \sf{ =g(f(x))}$

$\displaystyle \sf{ = {( {x}^{3}) }^{ \frac{1}{3} } }$

$= \sf{x}$

Hence f(x) & g(x) are inverse of one another

Hence proved

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