From the description, we have the function [tex]f(x)= \sqrt{ \frac{1}{2x-10}+3 } [/tex]
Since the square root cannot be a negative number, the only thing need to do to find the domain of the function [tex]f(x)[/tex] is take the expression inside the square rot and set it greater or equal than zero:
[tex]\frac{1}{2x-10}+3 \geq 0[/tex]
[tex] \frac{6x-29}{2x-10} \geq 0[/tex]
[tex] \frac{6x-29}{2(x-5)} \geq 0[/tex]
[tex]x \leq \frac{29}{6} [/tex] or [tex]x\ \textgreater \ 5[/tex]
We can conclude that we can use the inequality [tex]\frac{1}{2x-10}+3 \geq 0[/tex] to find the domain of [tex]f(x)[/tex]. Also, the domain of [tex]f(x)[/tex] is (∞,[tex] \frac{29}{6} [/tex]]U(5,∞).
