Find the domain of the following function. (x – 6) (x – 4) f(x) = x + 4 If the domain contains more than one interval, use the union symbol (U) in your answer.

Given function is $f\left(x\right)=\sqrt{\frac{\left(x-6\right)\left(x-4\right)}{x+4}}=\frac{\sqrt{\left(x-6\right)\left(x-4\right)}}{\sqrt{x+4}}$

the denominator of the function becomes zero when,

$$\begin{gathered} x+4=0 \\ x=-4 \end{gathered}$$

Thus, the function $f\left(x\right)$ approaches infinity when x approaches $-4$, and it doesn't have a defined value at $x=-4$. Thus, the point $x=-4$ is not included in its domain.

The function will give imaginary value if,

$$\begin{gathered} \left(x-6\right)\left(x-4\right)<0 \Rightarrow x\in \left(4,6\right) \\ \text{or}, x+4<0\Rightarrow x\in \left(-\infty ,-4\right) \end{gathered}$$

Thus, above interval are also not included in the domain of $f\left(x\right)$. Thus, in order to get the actual domain of the function $f\left(x\right)$, we have to subtract those intervals from the interval$\left(-\infty ,\infty \right)$.