Find the domain and range of f(x)=1/4x+11 . Write the domain and range in interval form, using inequalities, and in set notation.

1 answer

To find the domain and range of the function \( f(x) = \frac{1}{4}x + 11 \), we will analyze the function step by step.

Domain

The function \( f(x) = \frac{1}{4}x + 11 \) is a linear function. A linear function is defined for all real numbers, meaning there are no restrictions on the values that \( x \) can take.

In interval notation: \[ (-\infty, \infty) \]

In set notation: \[ { x \in \mathbb{R} } \]

Range

The output of a linear function can take any real value. The slope of the function is \(\frac{1}{4}\), and since the function is not bounded above or below, it will cover all real numbers as \( x \) varies.

In interval notation: \[ (-\infty, \infty) \]

In set notation: \[ { f(x) \in \mathbb{R} } \]

Summary

  • Domain:

    • Interval form: \( (-\infty, \infty) \)
    • Set notation: \( { x \in \mathbb{R} } \)
  • Range:

    • Interval form: \( (-\infty, \infty) \)
    • Set notation: \( { f(x) \in \mathbb{R} } \)

Thus, both the domain and range of the function \( f(x) = \frac{1}{4}x + 11 \) are \( (-\infty, \infty) \).