Question

If
${e}^{x} + {e}^{f(x)} = e \:$
Then find the range of the function f(x)​

Answer 1

.......................

Answer 2

Answer:

See below.

Step-by-step explanation:

The above function could be written as,

${e}^{f(x)} = e - {e}^{x}$

In the logarithmic form,

$log_{e}(e - {e}^{x} ) = f(x)$

Now, the above function is defined for values of,

$e - {e}^{x} > 0$

On solving,

$e > {e}^{x} \\ log_{e}(e) > log_{e}( {e}^{x} ) \\ 1 > x$

Hence, the domain of the function is the set of all real numbers smaller than 1.

In a simple form,

range = { x | x = log base=e ( e - e^a ) where 1 > a }