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**Illustration 1: Let f(x) = Î±x / (x+1), x â‰ -1. Then, for what value of Î± is f(f(x) = x? (2001)****Solution: **The given function f(x) = Î±x / (x+1), x â‰ -1

**Illustration 2: Suppose, f(x) = (x+1) ^{2} for x â‰¥ -1. If g(x) is the function whose graph is a reflection of the graph of f(x) with respect to the line y = x, then what is the value of g (x)? (2002)**

Let us assume y = f(x) = (x+1)

Then Â± âˆšy = (x+1) for x â‰¥ -1

Hence, âˆšy = x + 1 which gives y â‰¥ 0 and x+1 â‰¥ 0.

Hence, x = âˆšy - 1

So, f

Hence, f

**Illustration 3: Let f(x) = x ^{2} and g(x) = sin x for all x âˆˆ R. Then, the set of all x satisfying (fogogof)(x) = (gogof)(x), where (fog)(x) = f(g(x)), is (2011)**

Then (gof) (x) = sin x

Hence, go (gof) (x) = sin (sin x

Therefore, (fogogof) (x) = (sin (sin x

Again, (gof) (x) = sin x

So, (gogof) (x) = sin (sin x

Given, (fogogof) (x) = (gogof) (x)

Also, (sin (sin x

Hence, we get, sin (sin x

This gives either sin (sin x

Hence, sin x

Therefore, x

So, x = Â± âˆš nÏ€.

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