Find the Domain

Problem

y=√(,ln((4−x)/(x−2)))

Solution

1. Identify the constraints for the square root function, which requires the radicand to be non-negative:

ln((4−x)/(x−2))≥0

2. Identify the constraints for the natural log function, which requires the argument to be strictly positive:

(4−x)/(x−2)>0

3. Solve the inequality for the square root constraint by exponentiating both sides with base e

eln((4−x)/(x−2))≥e0

(4−x)/(x−2)≥1

4. Rearrange the inequality to compare it to zero:

(4−x)/(x−2)−1≥0

(4−x−(x−2))/(x−2)≥0

(6−2*x)/(x−2)≥0

5. Determine the critical points of the rational expression, which are x=3 (where the numerator is zero) and x=2 (where the denominator is zero).

6. Test the intervals (−∞,2) (2,3] and (3,∞) to see where the expression is non-negative. Note that x=2 is excluded because it makes the denominator zero.

For *x=2.5:(6−5)/(2.5−2)=1/0.5=2>0

For *x=4:(6−8)/(4−2)=(−2)/2=−1<0

7. Verify that the interval (2,3] also satisfies the natural log constraint (4−x)/(x−2)>0 Since 1>0 any value that makes the expression ≥1 automatically makes it >0

Final Answer

Domain: *(2,3]

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