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Need solution for RD Sharma Maths Class 12 Chapter Function Exercise 2.2 Question 10.

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Answer : h\; o(g\; o\; f)=(h\; o\; g)o\; f

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: N \rightarrow N, g: N \rightarrow N, h: N \rightarrow R defined as

             \begin{array}{r} f(x)=2 x \\ g(y)=3 y+4 \\ h(z)=\sin z \\ \text { All } x, y, z \in N \end{array}

Solution :

             \begin{aligned} h \circ(g \circ f)(x)=& h(g \circ f(x))=h(g(f(x))) \\ &=h(g(2 x))=h(3(2 x)+4) \\ &=h(6 x+4) \\ &=\sin (6 x+4) \quad \forall x \in N \end{aligned}


\begin{aligned} ((h \circ g) \circ f)(x) &=(h \circ g)(f(x)) \\ &=(h \circ g)(2 x) \\ &=h(g(2 x))=h(3(2 x)+4) \end{aligned}

                                 =h(6x+4)=\sin(6x+4) \; \; \; \; \; \; \; \forall x\; \epsilon \; N

This shows,   h\; o(g\; o\; f)=(h\; o\; g)o\; f

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