The numerator of the function is a square root expression. Because any positive number and zero can have a square root, so the number inside the square root must be
large and equal to zero. That is, 5 - |x - 3| >= 0. This is our first inequality. Look the denominator lg(8 - 2^{x}), one of the property of logarithm is: negative
number and zero do not have logarithm, so we get 8 - 2^{x} > 0. This is our second inequality. If the denominator of a fraction is zero, then the fraction will be undefined.
So the denominator of a fraction can not be zero. Another property of the logarithm is: the logarithm of one is zero. That is, log 1 = 0. So 8 - 2^{x} can not be one.
That is, 8 - 2^{x} not equal to 1. This is our third condition. The domain of the function must satisfy the three conditions. Solving the first inequality involved
in absolute value inequality. 5 - |x - 3| >= 0. Move the -|x - 3| to the right side of the inequality. We get |x - 3| <= 5. Apply the property of the absolute value
inequality, -5 <= x - 3 <= 5. Add 3 to each side of the inequalities, -2 <= x <= 8. So x must be in the range of [-2, 8]. In second inequality,
8 - 2^{x} < 0. Move the -2^{x} to the right side of the inequality, we get 2^{x} < 8. Note: 8 is equal to 2^{3}. So 2^{x} < 2^{3}.
Then x < 3. Combine the first inequality and the second inequality, we get that x must be in the range of [-2, 3]. Look the third condition: 8 - 2^{x} not equal to 1.
Move 1 to the left side of the not equal sign and move -2^{x} to the right side of the not equal sign. We get that 8 - 1 not equal to 2^{x}. That is, 7 not equal to 2^{x}.
Apply logarithm, then log 7 not equal to x log2, then x not equal to log7 ÷ log2. So the domain of y is its independent variable x in the range of [-2, 3] and x not equal to log7 ÷ log2.