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Find gof and fog when f: R → R and g : R → R is defined by

    \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{8}{{\mathbf{x}}^{\mathbf{3}}}~\]

and

    \[\mathbf{g}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{1}/\mathbf{3}}}\]

CBSE | Class 12 | Exercise 2.2 | Function | Maths | RD Sharma
Find gof and fog when f: R → R and g : R → R is defined by

    \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{8}{{\mathbf{x}}^{\mathbf{3}}}~\]

and

    \[\mathbf{g}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{1}/\mathbf{3}}}\]

Given, f: R → R and g: R → R

So, gof: R → R and fog: R → R

    \[f\left( x \right)\text{ }=\text{ }8{{x}^{3}}~\]

and 

    \[g\left( x \right)\text{ }=~{{x}^{1/3}}\]

(gof) (x) = g (f (x))

= 

    \[g~(8{{x}^{3}})\]

=

    \[{{(8{{x}^{3}})}^{1/3}}\]

=

    \[{{[{{\left( 2x \right)}^{3}}]}^{1/3}}\]

= 

    \[2x\]

Now, (fog) (x) = f (g (x))

    \[=\text{ }f\text{ }({{x}^{1/3}})\]

    \[=\text{ }8\text{ }{{({{x}^{1/3}})}^{3}}\]

    \[=~8x\]

i

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