domain of g(x)=(sqrt(x-2))/(x-7)
We save the definition: g(x) = x -2x -7
Let's find the domain of x -2x -7
For the expression x -2x -7 to exist, you must check: x -7 ≠ 0
For the expression x -2 to be defined, we must have: x -2 >= 0
Let's solve in ℝ: x -7 ≠ 0
Solution obtained: x ≠ 7
Let's solve in ℝ: x -2 >= 0
Solution found: x >= 2
Let's aggregate the different solutions to find the overall solution:
x-∞ 2 7 +∞
x -7 ≠ 0✅✅✅❌✅
x -2 >= 0❌✅✅✅✅
Domain of definition: ❌✅✅❌✅
▶Domain of definition:   x ∈ [2, 7) ∪ (7, +∞)