Select the correct domain for the rational function \( f(x)=\frac{3}{x-4} \) \( \begin{array}{l}x \text {. } \\ x<4 \\ x>4 \\ x \neq 0\end{array} \)

To determine the correct domain for the rational function \( f(x) = \frac{3}{x-4} \), we need to identify the values of \( x \) that make the function undefined. A rational function is undefined when its denominator is equal to zero. 1. **Identify the denominator**: The denominator of the function is \( x - 4 \). 2. **Set the denominator to zero**: \[ x - 4 = 0 \] Solving for \( x \) gives: \[ x = 4 \] 3. **Determine the domain**: The function is undefined at \( x = 4 \). Therefore, the domain of the function includes all real numbers except \( x = 4 \). In interval notation, the domain can be expressed as: \[ (-\infty, 4) \cup (4, \infty) \] Now, let's evaluate the options provided: - \( x \) (not correct, as it includes \( x = 4 \)) - \( x < 4 \) (not correct, as it excludes values greater than 4) - \( x > 4 \) (not correct, as it excludes values less than 4) - \( x \neq 0 \) (not correct, as it does not address the restriction at \( x = 4 \)) None of the provided options correctly represent the domain of the function. The correct domain is \( x \neq 4 \).