Question

Find the domain and range of f(x)=1/4x+11 in interval form, using inequalities, and in set notation.
Domain:Interval Form ( ) Inequalities ( ) Set Notation ( )
Range:Interval Form( ) Inequalities ( ) Set Notation( )

Answer 1

To analyze the function \( f(x) = \frac{1}{4}x + 11 \), we can determine its domain and range.

### Domain

**Interval Form**: The function \( f(x) \) is a linear function, which is defined for all real numbers. Thus, the domain in interval form is:
\[
(-\infty, \infty)
\]

**Inequalities**: The inequalities that describe the domain can be written as:
\[
x \in (-\infty, \infty)
\]

**Set Notation**: In set notation, the domain can be expressed as:
\[
\{ x \in \mathbb{R} \}
\]

### Range

**Interval Form**: As a linear function with a non-zero slope, the range is also all real numbers. Thus, the range in interval form is:
\[
(-\infty, \infty)
\]

**Inequalities**: For the range, the inequalities can be described as:
\[
f(x) \in (-\infty, \infty)
\]

**Set Notation**: In set notation, the range can be expressed as:
\[
\{ y \in \mathbb{R} \}
\]

### Summary

**Domain:**
- Interval Form: \((- \infty, \infty)\)
- Inequalities: \(x \in (-\infty, \infty)\)
- Set Notation: \(\{ x \in \mathbb{R} \}\)

**Range:**
- Interval Form: \((- \infty, \infty)\)
- Inequalities: \(f(x) \in (-\infty, \infty)\)
- Set Notation: \(\{ y \in \mathbb{R} \}\)