Functions-4

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Let f : R → R be such that f(x) = 2x. Determine (a) Range of f (b) {x : f(x) = 1}

(c) whether f(x+y) = f(x).f(y) holds.

Solution: (a) Since 2x is positive for every x ∈ R.

So, f(x) = 2x is a positive real number for every x ∈ R.

Moreover, for every positive real number x, there exist log2 x ∈ R such that f(log2 x) = 2log2x = x

Hence, we conclude that the range of f is the set of all positive real numbers.

(b) Since, f(x) = 1 => 2x = 1 => 2x = 20 => x = 0

Therefore, {x : f(x) = 1} = {0}.

(c) Since, f(x + y) = 2x + y = 2x 2y = f(x) f(y)

Therefore, f(x + y) = f(x) f(y) holds for all x, y ∈ R.

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