Find the domain of the function \( f(x)=\frac{2}{x-4} \) \[ \begin{array}{l}\{x \mid x>4\} \\ \{x \mid x<4\} \\ \{x \mid x \neq 4\}\end{array} \]

To find the domain of the function \( f(x) = \frac{2}{x-4} \), we need to identify the values of \( x \) that will not make the denominator zero. The expression \( x - 4 = 0 \) leads us to \( x = 4 \). Therefore, the function is undefined at this point. Thus, the domain of the function is all real numbers except for 4, which can be expressed as \( \{x \mid x \neq 4\} \). Now, let's consider some interesting tidbits! The function \( f(x) \) exhibits a vertical asymptote at \( x = 4 \). This means as \( x \) approaches 4 from either side, the value of \( f(x) \) tends to infinity or negative infinity, giving the graph its characteristic shape around that point. Additionally, the domain definition is crucial in real-world applications, especially in fields like engineering or physics. For example, if this function represents the behavior of a system, avoiding the value of 4 becomes necessary to ensure stability and avoid breakdowns in simulations or calculations.