Answer:
[tex](-\infty,-5)\cup(4,\infty)[/tex]
Explanation:
Given the below function;
[tex]\log (\frac{x+5}{x-4})[/tex]
Recall that the domain of a function is a set of input values for which the function is defined.
Note that a logarithmic function is only defined when the input is positive, so the given function is defined when x + 5 > 0 and x - 4 > 0.
Let's go ahead and solve the inequalities as seen below;
[tex]\begin{gathered} x+5>0 \\ x<-5 \\ \text{And } \\ x-4>0 \\ x>4 \end{gathered}[/tex]
Notice that for the function to be defined also, the denominator must not be equal to zero, so we'll have;
[tex]\begin{gathered} x-4\ne0 \\ x\ne4 \end{gathered}[/tex]
We can see that for the given function to be defined, the following restrictions have to be considered;
[tex]\begin{gathered} x<-5 \\ x>4 \\ x\ne4 \end{gathered}[/tex]
Therefore the domain of the given function can be written in interval notation as;
[tex](-\infty,-5)\cup(4,\infty)[/tex]
