If the function f - R - R be given by f x - x 2-2 and g - R - R be given by g x - x - x -1- y -1-find fog and gof and hence fing fog2 and gof-3-

We have fog(x)=f[g(x)]=f ( x/x 1 )= ( x/x 1 )^2+2 And, fog(2)= ( 2/2 1 )^2+2=6 Also, gof (x)=g[f(x)]=g(x^2+2)=x^2+2/x^2+2 1=x^2+2/x^2+1 And, gof( 3)=( 3)^2+2/( ...

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Standard XII

Mathematics

Sum of Binomial Coefficients of Even Numbered Terms

If the functi...

Question

If the function $f : R \to R$ be given by $f (x) = x^{2} + 2$ and $g : R \to R$ be given by $g (x) = \frac{x}{x - 1}, x \neq 1$ ,
find fog and gof and hence fing fog(2) and gof(-3).

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Solution

We have $f o g (x) = f [g (x)] = f (\frac{x}{x - 1}) = {(\frac{x}{x - 1})}^{2} + 2$

And, $f o g (2) = {(\frac{2}{2 - 1})}^{2} + 2 = 6$

Also, $g o f (x) = g [f (x)] = g (x^{2} + 2) = \frac{x^{2} + 2}{x^{2} + 2 - 1} = \frac{x^{2} + 2}{x^{2} + 1}$

And, $g o f (- 3) = \frac{(- 3)^{2} + 2}{(- 3)^{2} + 1} = \frac{11}{10}$

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If the function f:R→R be given by f(x)=x2+2 and g:R→R be given by g(x)=xx−1,x≠1, find fog and gof and hence fing fog(2) and gof(-3).

We have fog(x)=f[g(x)]=f(xx−1)=(xx−1)2+2 And, fog(2)=(22−1)2+2=6 Also, gof(x)=g[f(x)]=g(x2+2)=x2+2x2+2−1=x2+2x2+1 And, gof(−3)=(−3)2+2(−3)2+1=1110

If the function $f : R \to R$ be given by $f (x) = x^{2} + 2$ and $g : R \to R$ be given by $g (x) = \frac{x}{x - 1}, x \neq 1$ ,
find fog and gof and hence fing fog(2) and gof(-3).