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

Apr 20, 2015

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 ${x}^{2} - 7 x + 6$ cannot be zero (You cannot divide by zero).

From first condition you get an inequality $x - 5 \ge 0$ which has a solution x in (5; +oo)

Second condition leads to solving a square equation ${x}^{2} - 7 x + 6 \ne 0$
For this equation $\Delta = {7}^{2} - 4 \cdot 1 \cdot 6 = 25$
$\sqrt{\Delta} = 5$
x_1=2; x_2=6

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