Find the domain of the function `f(x)=(1)/(8{x}^(2)-6{x}+1).`

Explanation

To find the domain of the function \( f(x) = \frac{1}{8\{x\}^2 - 6\{x\} + 1} \), we need to determine the values of \( x \) for which the function is defined. The function will be undefined when the denominator is equal to zero.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the function is \( 8\{x\}^2 - 6\{x\} + 1 \). We need to find when this expression equals zero.

2. **Set the Denominator to Zero**:
\[ 8\{x\}^2 - 6\{x\} + 1 = 0 \]

3. **Use the Quadratic Formula**:
The quadratic formula is given by:
\[ \{x\} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 8 \), \( b = -6 \), and \( c = 1 \).
Substituting these values into the formula:
\[ \{x\} = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 8 \cdot 1}}{2 \cdot 8} \]
\[ = \frac{6 \pm \sqrt{36 - 32}}{16} \]
\[ = \frac{6 \pm \sqrt{4}}{16} \]
\[ = \frac{6 \pm 2}{16} \]

4. **Calculate the Two Possible Values**:
- For the positive root:
\[ \{x\} = \frac{8}{16} = \frac{1}{2} \]
- For the negative root:
\[ \{x\} = \frac{4}{16} = \frac{1}{4} \]

5. **Determine the Corresponding Values of \( x \)**:
The fractional part \( \{x\} \) can be expressed as:
\[ \{x\} = x - \lfloor x \rfloor \]
Therefore, if \( \{x\} = \frac{1}{2} \), then:
\[ x = n + \frac{1}{2} \quad \text{for } n \in \mathbb{Z} \]
If \( \{x\} = \frac{1}{4} \), then:
\[ x = n + \frac{1}{4} \quad \text{for } n \in \mathbb{Z} \]

6. **Identify the Points to Exclude from the Domain**:
The function is undefined at:
\[ x = n + \frac{1}{2} \quad \text{and} \quad x = n + \frac{1}{4} \quad \text{for } n \in \mathbb{Z} \]

7. **State the Domain**:
The domain of the function \( f(x) \) is all real numbers except for the points where the denominator is zero:
\[ \text{Domain of } f = \mathbb{R} \setminus \left\{ n + \frac{1}{2}, n + \frac{1}{4} \mid n \in \mathbb{Z} \right\} \]