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

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