Question

Let \[f:R \to R \] be such that \[f \left( x \right) = {2^x} \] . Determine
i.Range of \[f \] .
ii. \[ \left \{ {x:f \left( x \right) = 1} \right \} \]
iii.Whether \[f \left( {x + y} \right) = f \left( x \right)f \left( y \right) \] holds.

Answer 1

Hint: In this question a function is given which is real for all real values of \[f \] , so as per each three conditions given we will find the range of the function, its value when function \[f \left( x \right) = 1 \] and weather \[f \left( {x + y} \right) = f \left( x \right)f \left( y \right) \] is true.

Complete step-by-step answer:
Given the function \[f \left( x \right) = {2^x} \] and the value of this function is always real
Now for the range of the function \[f \left( x \right) = {2^x} \] , since the value function has to always be real value so we will substitute the values of x as positive real numbers since negative values will not give a real value.
So the range of the function for real values should be \[{R^ + } \] or \[ \left( {0, \infty } \right) \]
For \[ \left \{ {x:f \left( x \right) = 1} \right \} \]
Now when the value of the function is 1
Since the given function is \[f \left( x \right) = {2^x} \] , hence we can write
 \[{2^x} = 1 \]
We can also write \[1 = {2^0} \] , since any number to the power 0 is always 1, hence we can further write the function as
 \[\Rightarrow {2^x} = {2^0} \]
Now as we know if the base of the exponential power is same on both the sides then we can equate its power, hence we can write
 \[\Rightarrow x = 0 \]
 \[\Rightarrow f \left( {x + y} \right) = f \left( x \right)f \left( y \right) \]
Now to check the function \[f \left( {x + y} \right) = f \left( x \right)f \left( y \right) \]
Substitute \[x \to x + y \] in the given function as
 \[\Rightarrow f \left( {x + y} \right) = {2^{x + y}} \]
Now as we know the multiple of two exponents with the same power is the sum of the power for the two exponents \[{a^m}.{a^n} = {a^{m + n}} \] , hence we can write
 \[\Rightarrow f \left( {x + y} \right) = {2^x}{.2^y} \]
Hence we can say the function \[f \left( {x + y} \right) = f \left( x \right)f \left( y \right) \]

Note: Range of a function is defined as the value of the function which is defined for each value of the parameter available in the domain. For example, $f(x) = {x^2}$ is a function such that the domain is $ \left \{ {1,2,3} \right \}$ then, the range of the function f(x) is defined as $ \{ {1^2},{2^2},{3^2} \} = \{ 1,4,9 \} $ .