Question
Find the domain of the function \( f(x)=\sqrt{21-3 x} \) What is the domain of \( f \) ? \( \square \) (Type your answer in interval notation.)
Ask by Mcguire Hodges. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The domain of \( f \) is all real numbers less than or equal to 7, written as \((-\infty, 7]\).
Solution
To find the domain of the function \( f(x) = \sqrt{21 - 3x} \), we need to ensure that the expression inside the square root is non-negative:
\[
21 - 3x \geq 0
\]
Solving for \( x \):
\[
-3x \geq -21
\]
\[
x \leq 7
\]
Therefore, the domain of \( f \) consists of all real numbers \( x \) such that \( x \) is less than or equal to 7.
**Domain of \( f \):** \((-\infty, 7]\)
\[\boxed{\,(-\infty,\ 7]\,}\]
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Simplify this solution Extra Insights
To find the domain of the function \( f(x) = \sqrt{21 - 3x} \), we need to ensure that the expression inside the square root is non-negative. This means we set up the inequality: \[ 21 - 3x \geq 0. \] Solving for \( x \), we get: \[ 21 \geq 3x \] or \[ 7 \geq x, \] which can be rewritten as \[ x \leq 7. \] Since there's no restriction on \( x \) being negative or positive from the square root requirement, we can include all values up to 7. Therefore, the domain in interval notation is: \[ (-\infty, 7]. \]
