Q. 165.0( 1 Vote )

# Let f : R <span l

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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