Find the Domain of \(f(x)=\sqrt{\frac{x-2}{3-x}}\)

Solution

Given: $$ f(x)=\sqrt{\frac{x-2}{3-x}} $$

For a square root function, the expression inside the root must be non-negative.

Therefore, $$ \frac{x-2}{3-x}\ge0 $$

Critical values are: $$ x=2,\quad x=3 $$

Check the sign of $$ \frac{x-2}{3-x} $$ in the intervals:

$$ (-\infty,2),\quad (2,3),\quad (3,\infty) $$

For $$ x<2, $$ the expression is negative.

For $$ 2

For $$ x>3, $$ the expression is negative.

Also, $$ x=2 $$ is allowed since the value becomes \(0\).

But $$ x=3 $$ is not allowed because the denominator becomes zero.

Hence, the domain is: $$ [2,3) $$