To find the domain of the function [tex]\( f(x) = \frac{7x}{x-3} \)[/tex], we need to determine the values of [tex]\( x \)[/tex] that make the function undefined. This function is a rational function, and rational functions are undefined where the denominator is zero.
Here are the steps to solve this problem in detail:
1. Identify the Denominator:
The denominator of the given function is [tex]\( x - 3 \)[/tex].
2. Set the Denominator Equal to Zero:
To find the values that make the function undefined, set the denominator equal to zero:
[tex]\[
x - 3 = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Solving the equation [tex]\( x - 3 = 0 \)[/tex] gives:
[tex]\[
x = 3
\][/tex]
4. Determine the Domain:
Since the function is undefined when [tex]\( x = 3 \)[/tex], the domain of [tex]\( f(x) = \frac{7x}{x-3} \)[/tex] includes all real numbers except [tex]\( x = 3 \)[/tex].
Hence, the domain of [tex]\( f(x) \)[/tex] is all real numbers except [tex]\( 3 \)[/tex].
Therefore, the correct answer is:
All real numbers except 3.
