To determine the domain of the function [tex]\( f(x) = \frac{3x}{x-1} \)[/tex], we need to identify all the values of [tex]\( x \)[/tex] for which the function is defined. The function [tex]\( f(x) \)[/tex] will be undefined if the denominator is zero because division by zero is undefined in mathematics.
Let's look at the denominator of the function:
[tex]\[ x - 1 \][/tex]
We set the denominator equal to zero to find the values that would make the function undefined:
[tex]\[ x - 1 = 0 \][/tex]
Solving this equation gives:
[tex]\[ x = 1 \][/tex]
This tells us that the function is undefined when [tex]\( x = 1 \)[/tex]. Therefore, the function [tex]\( f(x) \)[/tex] is defined for all real numbers except [tex]\( x = 1 \)[/tex].
Thus, the domain of [tex]\( f(x) = \frac{3x}{x-1} \)[/tex] is all real numbers except 1.
So, the correct answer is:
All real numbers except 1.
