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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)=x/(x−1), x≠1, find fog and gof and hence find fog (2) and gof (−3). - Mathematics

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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)=x/(x−1)` , x1, find fog and gof and hence find fog (2) and gof (−3).

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Solution

Here, f[x] = x2 + 2 and g(x)=x/(x1), x1

fog (x)=g2(x)+2
`⇒fog(x)=x^2/(x−1)^2+2`

`⇒fog (x)=(x^2+2(x−1)^2)/(x−1)^2`

Now, `fog (2)=(2^2+2(2 − 1)^2)/(2−1)^2=(4+2)/1=6`

Similarly, `gof (x)=(x^2+2)/(x^2+2−1)=(x^2+2)/(x^2+1)`

`⇒gof (−3)=((−3)^2+2)/((−3)^2+1)=11/10`

Concept: Inverse of a Function
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