Question

How do you find the domain of `f(x) = (sqrt[x - 5](x - 6))/(x^2 - 7x + 6)`?

Answer 1

The answer is : `D = (5;6) uu (6;oo)`

The domain of a function is this subset of real numbers for which all the operations in the functions have sense.

In this case
1) the expression `x-5` must be greater than or equal to zero (the square roots of negative numbers are not real)
2) the expression in denominator `x^2-7x+6` cannot be zero (You cannot divide by zero).

From first condition you get an inequality `x-5 >=0` which has a solution `x in (5; +oo)`

Second condition leads to solving a square equation `x^2-7x+6!=0`
For this equation `Delta = 7^2 - 4*1*6 = 25`
`sqrt(Delta) = 5`
`x_1=2; x_2=6`

So the answer is `D = (5;6) uu (6;oo)`