Question

Choose the correct domain from the following options given below for the function$f\left( x \right) = \sqrt {1 + \ln \left( {1 - x} \right)}$

1. $- \infty < x \leqslant 0$
2. $- \infty < x \leqslant \dfrac{{e - 1}}{e}$
3. $x \geqslant 1 - e$
4. $- \infty < x \leqslant 1$

Answer 1

Now to find the domain for any function we need to divide the given function into sub functions for which the domains are known to us. After dividing the given function into sub functions for which the domains are known we have to replace the expression inside the function and find the domain with respect to a given function.

Complete step-by-step solution:
Given,
$f\left( x \right) = \sqrt {1 + \ln \left( {1 - x} \right)}$,
Let us define,
$g\left( y \right) = \sqrt y$,
$h\left( z \right) = \ln \left( z \right)$.
We know the general domain of the functions $g,h$, it is$y \geqslant 0$,$z > 0$.
Now both functions $g,h$are in $f$ but in some indirect form.
Now in the place of $z$, $f$ has $1 - x$, so as the domain of $h$is $z > 0$ $\Rightarrow 1 - x > 0$ in $f$.
Therefore, $x < 1$.
There is another function $g$in $f$but in the place of $y$we have $1 + \ln \left( {1 - x} \right)$
Since the function $g$has a domain $y \geqslant 0$ $\Rightarrow 1 + \ln \left( {1 - x} \right) \geqslant 0$
$\Rightarrow \ln \left( {1 - x} \right) \geqslant - 1$,
By raising to the power of $e$on both sides of inequation, we get
$\Rightarrow 1 - x \geqslant \dfrac{1}{e}$,
$\Rightarrow x \leqslant 1 - \dfrac{1}{e}$.
Clearly,$\Rightarrow 1 \geqslant 1 - \dfrac{1}{e} \geqslant x$
Therefore the required domain is $- \infty \leqslant x \leqslant 1 - \dfrac{1}{e}$.
The correct option is 2, since $1 - \dfrac{1}{e} = \dfrac{{e - 1}}{e}$.

Note: In any function given to find the domain at first, we need to divide and rewrite the given function into the composition of the functions for which the domains are known or can be defined easily. In this problem, we have composed the given function $f$ in $goh$. Where the domains of $g,h$are known for us. After that, we have to use the domains of $g,h$ to determine the domain of a given function in the above shown manner. This is an ideal process to determine the domain of any given function.