Q. 165.0( 1 Vote )

Let f : R <span l

Answer :

Given that f: R R such that f(x) = 2x

To find: (i) Range of x


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


Moreover, for every positive real number x , log2x R such that



= x


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


To find: (ii) {x : f(x) = 1}


We have, f(x) = 1 …(a)


and f(x) = 2x …(b)


From eq. (a) and (b), we get


2x = 1


2x = 20 [ 20 = 1]


Comparing the powers of 2, we get


x = 0


{x : f(x) = 1} = {0}


To find: (iii) f(x + y) = f(x). f(y) for all x, y ϵ R


We have,


f(x + y) = 2x + y


= 2x.2y


[The exponent "product rule" tells us that, when multiplying two powers that have the same base, you can add the exponents or vice - versa]


= f(x).f(y) [f(x) = 2x]


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


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