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If `f:R -> R, g:R -> R` defined as `f(x) = sin x and g(x) = x^2`, then find the value of `(gof)(x) and (fog)(x) `and also prove that `gof != fog`.

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(fog) (x) = f[(g(x)]
= `f(x^(2)) = sin x^(2)`Ans.
and (gof)(x)= g[f(x)]
=g(sinx) = `(sinx)^(2)= sin^(2) x` Ans.
Therefore, fog `ne` gof . Hence Proved.
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NAGEEN PRAKASHAN-RELATIONS AND FUNCTIONS -Miscellaneous Exercise
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  6. Show that the function f: R->Rgiven by f(x)=x^3is injective.

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  7. Give examples of two functions f:" "N->Z" "a n dg:" "Z->Z such that o...

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  8. Given examples of two functions f:" "N ->N" "a n d""""""g:" "N->N such...

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  13. Consider the binary operations*: RxxR->R and o: RxxR->R defined as ...

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  14. Given a non -empty set X, let *:" "P(X)" "xx" "P(X) ->P(X) be defined ...

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  15. Define a binary operation * on the set A={0,1,2,3,4,5} as a*b=a+b (mod...

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  16. Let A" "=" "{-1," "0," "1," "2} , B" "=" "{-4," "-2," "0," "2} and f,g...

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  18. Let A = {1, 2, 3}. Then number of equivalence relations containing (1...

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  19. Let f: R->Rbe the Signum Function defined as f(x)={1,x >0 0,x=0-1,x<1 ...

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  20. Number of binary operations on the set {a, b} are (A) 10 (B) 16 (C)...

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