Question
4 of 5 What is the domain of \( y=3 \sqrt{-2 x-7}-2 \) ? Enter any fractions as improper fractions. (1 point) The domain of the function is ( \( \square, \square \). Check answer Remaining Attempts : 3
Ask by Bradley Bowers. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The domain of the function is all real numbers \( x \) where \( x \leq -\frac{7}{2} \).
Solution
To determine the domain of the function \( y = 3 \sqrt{-2x - 7} - 2 \), we need to ensure that the expression inside the square root is non-negative:
\[
-2x - 7 \geq 0
\]
Solving for \( x \):
\[
-2x \geq 7 \\
x \leq -\frac{7}{2}
\]
Therefore, the domain of the function is all real numbers \( x \) such that \( x \leq -\frac{7}{2} \).
**Domain:** \( (-\infty, -\frac{7}{2}] \)
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Simplify this solution Mind Expander
To find the domain of the function \( y=3 \sqrt{-2 x-7}-2 \), we start by identifying where the expression under the square root is non-negative, since the square root of a negative number is undefined in the real number system. Thus, we set up the inequality: \[ -2x - 7 \geq 0 \] Solving this gives: \[ -2x \geq 7 \implies x \leq -\frac{7}{2} \] We can conclude that the domain of the function is: \[ (-\infty, -\frac{7}{2}] \] So, the domain of the function is ( \( -\infty, -\frac{7}{2} \) ).
