Question

The domain of the function $f(x) = \log (1 - x) + \sqrt {{x^2} - 1} $
A. $( - \infty , - 1)$
B. $( - \infty , - 1)$
C. $( - \infty ,2)$
D. $( - \infty ,0)$

Answer 1

Hint: The domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. We should not get confused between domain and range. Range tells about the possible output values of the function.

Complete Step-by-Step solution:
Given,
$f(x) = \log (1 - x) + \sqrt {{x^2} - 1} $
For (1-x)
       $
  1 - x > 0 \\
  x < 1 \\
 $
For (${x^2} - 1$)
      $
  {x^2} - 1 \geqslant 0 \\
  (x - 1)(x + 1) \geqslant 0 \\
  x \in ( - \infty , - 1] \cup [1,\infty ) \\
$
Therefore,$x \in ( - \infty , - 1)$.

Note: A function with a fraction with a variable in the denominator. To find the domain of function, set the bottom equal to zero and exclude the x value you find when you solve the equation. Whereas range can be calculated by putting all possible values of input that is domain of the function.