Find the domain of the following function.
$y=\frac{2x+3}{x-4}$

• A. $(-\infty ,\infty )$
• B. $(-\infty ,4)\cup (4,\infty )$
• C. $(-\infty ,4]$
• D. $[4,\infty )$

Answer: (B)

Verified step by step guidance

1. Identify the function given: \(y=\frac{2x+3}{x-4}\). This is a rational function, which means the domain includes all real numbers except where the denominator is zero.
2. Set the denominator equal to zero to find values that are not allowed in the domain: \(x - 4 = 0\).
3. Solve the equation for \(x\): \(x = 4\). This value makes the denominator zero, which is undefined in mathematics.
4. Therefore, the domain of the function is all real numbers except \(x = 4\). In interval notation, this means the domain is \((-\infty, 4) \cup (4, \infty)\).
5. Express the domain clearly: the function is defined for every real number except at \(x=4\), where the function has a vertical asymptote.