Find domain for
`" "y=cos^(-1)((1-2abs(x))/3)+log_abs(x-1)x.`

Answer

To find the domain of the function \( y = \cos^{-1}\left(\frac{1 - 2|x|}{3}\right) + \log_{|x|}(x - 1) \), we need to determine the domain of each component of the function separately and then find their intersection.

### Step 1: Determine the domain of \( f_1(x) = \cos^{-1}\left(\frac{1 - 2|x|}{3}\right) \)

The function \( \cos^{-1}(z) \) is defined for \( z \) in the interval \([-1, 1]\). Therefore, we need to solve the following inequality:

\[ -1 \leq \frac{1 - 2|x|}{3} \leq 1 \]

#### Step 1.1: Solve the left inequality

\[ -1 \leq \frac{1 - 2|x|}{3} \]

Multiplying all sides by 3:

\[ -3 \leq 1 - 2|x| \]

Subtracting 1 from both sides:

\[ -4 \leq -2|x| \]

Dividing by -2 (remember to reverse the inequality):

\[ 2 \geq |x| \]

This implies:

\[ |x| \leq 2 \]

Thus, we have:

\[ -2 \leq x \leq 2 \]

#### Step 1.2: Solve the right inequality

\[ \frac{1 - 2|x|}{3} \leq 1 \]

Multiplying all sides by 3:

\[ 1 - 2|x| \leq 3 \]

Subtracting 1 from both sides:

\[ -2|x| \leq 2 \]

Dividing by -2 (again, reversing the inequality):

\[ |x| \geq -1 \]

Since \( |x| \) is always non-negative, this condition is always satisfied.

### Conclusion for \( f_1(x) \)

The domain of \( f_1(x) \) is:

\[ [-2, 2] \]

### Step 2: Determine the domain of \( f_2(x) = \log_{|x|}(x - 1) \)

For the logarithmic function to be defined, we need two conditions:

1. The base \( |x| \) must be greater than 0 and not equal to 1.
2. The argument \( x - 1 \) must be greater than 0.

#### Step 2.1: Base condition

\(|x| > 0\) implies \( x \neq 0 \).

\(|x| \neq 1\) implies \( x \neq 1 \) and \( x \neq -1 \).

#### Step 2.2: Argument condition

\[ x - 1 > 0 \implies x > 1 \]

### Conclusion for \( f_2(x) \)

The domain of \( f_2(x) \) is:

\[ (1, \infty) \quad \text{(excluding } x = 0, 1, -1\text{)} \]

### Step 3: Find the intersection of the domains

Now, we need to find the intersection of the domains of \( f_1 \) and \( f_2 \):

- Domain of \( f_1 \): \( [-2, 2] \)
- Domain of \( f_2 \): \( (1, \infty) \)

The intersection is:

\[ (1, 2] \]

### Final Domain

Thus, the domain of the function \( y \) is:

\[ \boxed{(1, 2]} \]