Question

use set builder interval notation to express domain and range of the function: f(x) = 3x + 4

Answer 1

To express the domain and range of the function \( f(x) = 3x + 4 \) using set builder notation and interval notation, we will analyze the function.

### Domain:
The function \( f(x) = 3x + 4 \) is a linear function, which means it is defined for all real numbers. Therefore, the domain can be expressed as:

- **Set Builder Notation:**
\[
\{ x \in \mathbb{R} \}
\]

- **Interval Notation:**
\[
(-\infty, \infty)
\]

### Range:
For a linear function \( f(x) = mx + b \) (where \( m \neq 0 \)), the range is also all real numbers. The function \( f(x) = 3x + 4 \) will take all values from \(-\infty\) to \(\infty\).

- **Set Builder Notation:**
\[
\{ f(x) \in \mathbb{R} \}
\]

- **Interval Notation:**
\[
(-\infty, \infty)
\]

### Summary:
- **Domain:**
- Set Builder: \( \{ x \in \mathbb{R} \} \)
- Interval: \( (-\infty, \infty) \)

- **Range:**
- Set Builder: \( \{ f(x) \in \mathbb{R} \} \)
- Interval: \( (-\infty, \infty) \)

Thus, the domain and range of the function \( f(x) = 3x + 4 \) are both all real numbers.