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