What is the domain of \(f(x)=\log_{2}\left(\frac{x+4}{x-1}\right)\)?

Correct answer: \(( -\infty,-4)\cup(1,\infty)\)

For a logarithmic function, its argument must be strictly positive. Hence, \(\frac{x+4}{x-1}>0\) is required. The critical values are \(x=-4\) and \(x=1\). A sign check shows that the fraction is positive for \(x<-4\) and for \(x>1\). At \(x=-4\), the argument becomes 0, which is not allowed for a logarithm, and at \(x=1\), the fraction is undefined. Therefore, the domain is \(( -\infty,-4)\cup(1,\infty)\). Exam tip: In logarithmic questions, first impose that the complete argument is greater than zero.