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 consists of all the possible values of \( x \) for which the function is defined. In this case, the function is a rational function, which means we need to ensure that the denominator is not equal to zero.

1. **Set the denominator equal to zero:**
\[ 3x - 4 = 0 \]

2. **Solve for \( x \):**
\[ 3x = 4 \implies x = \frac{4}{3} \]

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

### Step 2: Determine the Range
To find the range, we need to express \( y \) in terms of \( x \) and analyze the resulting equation.

1. **Set \( f(x) \) equal to \( y \):**
\[ y = \frac{2 - 5x}{3x - 4} \]

2. **Cross-multiply to eliminate the fraction:**
\[ y(3x - 4) = 2 - 5x \]

3. **Expand and rearrange the equation:**
\[ 3xy - 4y = 2 - 5x \]
\[ 3xy + 5x = 4y + 2 \]

4. **Factor out \( x \):**
\[ x(3y + 5) = 4y + 2 \]

5. **Solve for \( x \):**
\[ x = \frac{4y + 2}{3y + 5} \]

6. **Determine when this expression is undefined:**
The expression for \( x \) will be undefined when the denominator is zero:
\[ 3y + 5 = 0 \]
\[ 3y = -5 \implies y = -\frac{5}{3} \]

7. **Conclusion for Range:**
The function can take all real values except \( y = -\frac{5}{3} \):
\[ \text{Range} = \mathbb{R} \setminus \left\{ -\frac{5}{3} \right\} \]

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