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Question
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).
Solution
Here, f[x] = x2 + 2 and g(x)=x/(x−1), x≠1
∴ 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`
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