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Provide solution for RD Sharma Maths Class 12 Chapter Functions Exercise 2.2 Question 1 Sub question (vi).



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Answer : g\; o\; f=2x\; \text {and}\; f\; o\; g=8x

Hint : g\; o\; f means f(x) function is in g(x) function

         f\; o\; g means g(x) function is in f(x) function

Given : f(x)=8x^{3}


Solution :

Since  f:R\rightarrow R \; \text {and} \; g:R\rightarrow R

          f\; o\; g:R\rightarrow R \; \text {and} \; g\; o\; f:R\rightarrow R

Now, g\; o\; f(x)=g(f(x))=g(8x^{3} )

          g\; o\; f(x)=(8x^{3} )^{\frac{1}{3}}

                             =\left ( 8 \right )^{\frac{1}{3}}=\left ( 2^{3}\right )^{\frac{1}{3}}                                                                       \because \left [ \text {as we know},\left ( a^{m} \right )^{\frac{1}{n}}=\left ( a \right )^{\frac{m}{n}} \right ]

       g\; o\; f (x)=(8)^{\frac{1}{3}}(x^{3})^{\frac{1}{3}}

       g\; o\; f (x)= 2x


f\; o\; g (x)=f(g(x))=f(x)^{\frac{1}{3}}

f\; o\; g (x)=8 (x^{\frac{1}{3}})^{3}

f\; o\; g (x)=8 x                      \because \left [ (a^{m})^{\frac{1}{n}}=(a)^{\frac{m}{n}} \right ]

Hence, f\; o\; g=8x\; \text {and}\; g\; o\; f=2x

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