sympy import symbols, sqrt, solve, S

Define the variable and the function

x = symbols('x') expression = sqrt((1 - x) / (x - 2) + 2)

Find the conditions for the domain

1. The argument inside the square root must be non-negative

non_negative_condition = solve((1 - x) / (x - 2) + 2 >= 0, x)

2. The denominator of the fraction must not be zero

non_zero_condition = x - 2

(non_negative_condition, non_zero_condition)To find the domain of the function $y = \sqrt{\frac{1-x}{x-2} + 2}$, we analyze two conditions:

1. Non-negativity of the square root argument: The expression inside the square root, $\frac{1-x}{x-2} + 2$, must be $\geq 0$.

   • Solving this inequality gives two intervals where the expression is valid: $x \in (-\infty, 2) \cup [3, \infty).$

2. Excluding division by zero: The denominator $x-2$ in the fraction must not be zero.

   • This excludes $x = 2$.

Domain:

The domain is the intersection of these conditions: $x \in (-\infty, 2) \cup [3, \infty).$

Would you like me to visualize this graphically or elaborate further?

Related Questions:

1. How do we find the range of this function given its domain?
2. How does the behavior of the function change near $x = 2$?
3. Can we rewrite the expression inside the square root in simpler terms?
4. How would this function behave if we replaced the square root with a cube root?
5. What happens if $+2$ outside the square root is changed to another constant?

Tip:

Always check for points where the denominator equals zero and square root arguments become negative—they define the function's domain limits.