Find the domain and range of `f(x)=(2-5x)/(3x-4)`.

Explanation

To find the domain and range of the function \( f(x) = \frac{2 - 5x}{3x - 4} \), we will follow these steps:

### Step 1: Determine the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, we need to ensure that the denominator is not equal to zero.

1. Set the denominator equal to zero and solve for \( x \):
\[ 3x - 4 = 0 \]
\[ 3x = 4 \]
\[ x = \frac{4}{3} \]

2. The function is undefined at \( x = \frac{4}{3} \). Therefore, the domain of \( f(x) \) is all real numbers except \( \frac{4}{3} \):
\[ \text{Domain} = \mathbb{R} - \left\{ \frac{4}{3} \right\} \]

### Step 2: Determine the Range
To find the range, we will express \( y \) in terms of \( x \) and then solve for \( x \) in terms of \( y \).

1. Start with the equation:
\[ y = \frac{2 - 5x}{3x - 4} \]

2. Rearranging gives:
\[ y(3x - 4) = 2 - 5x \]
\[ 3yx - 4y = 2 - 5x \]

3. Rearranging to isolate \( x \):
\[ 3yx + 5x = 2 + 4y \]
\[ x(3y + 5) = 2 + 4y \]
\[ x = \frac{2 + 4y}{3y + 5} \]

4. The function \( x \) will be undefined when the denominator is zero:
\[ 3y + 5 = 0 \]
\[ 3y = -5 \]
\[ y = -\frac{5}{3} \]

5. Therefore, the range of \( f(x) \) is all real numbers except \( -\frac{5}{3} \):
\[ \text{Range} = \mathbb{R} - \left\{ -\frac{5}{3} \right\} \]

### Final Answer
- **Domain:** \( \mathbb{R} - \left\{ \frac{4}{3} \right\} \)
- **Range:** \( \mathbb{R} - \left\{ -\frac{5}{3} \right\} \)