To determine the domain of the function [tex]\( f(x) = \frac{x^2 + 1}{x - 1} \)[/tex], we need to identify the values of [tex]\( x \)[/tex] for which the function is undefined. The function [tex]\( f(x) \)[/tex] becomes undefined when the denominator is zero because division by zero is not allowed in mathematics.
Let’s find the value of [tex]\( x \)[/tex] that makes the denominator zero:
[tex]\[
x - 1 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = 1
\][/tex]
So, the function [tex]\( f(x) = \frac{x^2 + 1}{x - 1} \)[/tex] is undefined at [tex]\( x = 1 \)[/tex].
The domain of [tex]\( f(x) \)[/tex] is therefore all real numbers except [tex]\( x = 1 \)[/tex].
This can be written as:
[tex]\[
R - \{1\}
\][/tex]
Thus, the correct answer is:
(iii) [tex]\( R - \{1\} \)[/tex]
