Question
Find the domain of the function using interval notation. \[ f(x)=\sqrt{12-11 x} \]
Ask by Owen Gross. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The domain of the function \( f(x)=\sqrt{12-11x} \) is all real numbers \( x \) where \( x \le \frac{12}{11} \), which in interval notation is \( (-\infty, \frac{12}{11}] \).
Solution
To find the domain of the function \( f(x)=\sqrt{12-11x} \), we need to determine the values of \( x \) for which the radicand (the expression inside the square root) is non-negative.
1. Set up the inequality:
\[
12-11x \ge 0
\]
2. Solve the inequality:
\[
12 \ge 11x \quad \Longrightarrow \quad x \le \frac{12}{11}
\]
3. Thus, the domain of the function is all real numbers \( x \) such that:
\[
x \le \frac{12}{11}
\]
4. In interval notation, the domain is:
\[
(-\infty, \frac{12}{11}]
\]
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Simplify this solution The Deep Dive
To determine the domain of the function \( f(x) = \sqrt{12 - 11x} \), we need to ensure that the expression under the square root is non-negative because the square root of a negative number is not defined in the real number system. This leads us to the inequality: \[ 12 - 11x \geq 0 \] Solving this gives: \[ 12 \geq 11x \] Dividing both sides by 11: \[ \frac{12}{11} \geq x \] or, equivalently: \[ x \leq \frac{12}{11} \] Thus, the domain of the function is all real numbers \( x \) that are less than or equal to \( \frac{12}{11} \). In interval notation, the domain is: \[ (-\infty, \frac{12}{11}] \]
