Domain of the function f(x) = √(ln(x^2 - 4x + 4)) to the base (|x|-1)

To find the domain of a function, we need to determine the values of x for which the function is defined. In this case, we have a composite function with a square root and a natural logarithm. Therefore, we need to consider the domains of both functions and any restrictions that may arise from the composition.

1. Domain of the Square Root Function:
The square root function is defined for non-negative real numbers. Thus, the expression inside the square root, ln(x^2 - 4x + 4), must be non-negative.

Since the natural logarithm is only defined for positive numbers, we have the following inequality:

ln(x^2 - 4x + 4) ≥ 0

To solve this inequality, we can consider the logarithmic property that states the logarithm of a product is equal to the sum of the logarithms of the individual factors:

ln((x - 2)^2) ≥ 0

Using the property that the logarithm of a number to the base e is equal to zero if and only if the number itself is equal to one, we have:

(x - 2)^2 = 1

Taking the square root of both sides, we get:

x - 2 = ±1

Solving for x, we have two possible solutions:

x - 2 = 1 → x = 3
x - 2 = -1 → x = 1

Therefore, the expression inside the square root, ln(x^2 - 4x + 4), is non-negative for x ∈ {1, 3}.

2. Domain of the Natural Logarithm Function:
The natural logarithm function, ln(x), is defined for positive real numbers. In this case, the base of the logarithm is |x|-1, which means the expression inside the absolute value must be positive.

|x| - 1 > 0

Solving this inequality, we have:

|x| > 1

This implies that x must be either greater than 1 or less than -1.

3. Composite Function:
To find the domain of the composite function, we need to consider the intersection of the domains of the square root function and the natural logarithm function.

From step 1, we found that the expression inside the square root, ln(x^2 - 4x + 4), is non-negative for x ∈ {1, 3}.

From step 2, we found that x must be either greater than 1 or less than -1.

Taking the intersection of these two sets, we find that the domain of the composite function f(x) = √(ln(x^2 - 4x + 4)) to the base (|x|-1) is:

Domain: x ∈ (-∞, -1) ∪ (1, 3)

Therefore, the function is defined for all real numbers except for x = -1, 0, 1, and 3.

(-2,-1) U (2, infinity) ??