Functions

To find the domain of the real-valued function f(x)=‌

1

√log0.5(2x−3)

+√4−9x2, we analyze the conditions under which each part of the function is defined.
First Part: ‌

1

√log0.5(2x−3)

For the square root in the denominator to be defined and non-zero, the expression inside must be positive: log0.5(2x−3)>0.
The base of the logarithm is less than 1 . This means:
2x−3<1‌‌⇒‌‌x<2
Additionally, the expression inside the logarithm must be positive:
2x−3>0‌‌⇒‌‌x>‌

3

2

Second Part: √4−9x2
The expression under the square root must be non-negative:
4−9x2>0‌‌⇒‌‌x2<‌

4

9

Solving for x, we derive:
−‌

2

3

<x<‌

2

3

Intersection of Conditions:
From the first part, we need x to satisfy both conditions: ‌

3

2

<x<2.
From the second part, x must be in the interval −‌

2

3

<x<‌

2

3

.
The intersection of these conditions is empty, as there is no number that simultaneously satisfies ‌

3

2

<x<2 and −‌

2

3

<x<‌

2

3

.
Hence, the domain of the function is a null set.