Question
Find the domain of \( \mathrm{f}(\mathrm{x}) \). \[ f(x)=\sqrt{-8-8 x} \] Answer Attempt 1 out of 2 Answer Type: Interval
Ask by Turnbull Willis. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The domain of \( f(x) = \sqrt{-8 - 8x} \) is all real numbers less than or equal to \(-1\), which is written as \( (-\infty, -1] \).
Solution
To determine the domain of the function \( f(x) = \sqrt{-8 - 8x} \), we need to ensure that the expression inside the square root is non-negative (since the square root of a negative number is not a real number).
### Step-by-Step Solution:
1. **Set the radicand (the expression inside the square root) greater than or equal to zero:**
\[
-8 - 8x \geq 0
\]
2. **Solve for \( x \):**
\[
-8x \geq 8
\]
\[
x \leq -1
\]
(Note: When dividing or multiplying by a negative number, the inequality sign flips.)
3. **Interpret the inequality:**
- The function \( f(x) \) is defined for all real numbers \( x \) that are less than or equal to \(-1\).
### Domain in Interval Notation:
\[
(-\infty, -1]
\]
**Final Answer:**
\[
\boxed{\,(-\infty,\ -1\,]\,}
\]
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Simplify this solution Extra Insights
To find the domain of \( f(x) = \sqrt{-8 - 8x} \), we need the expression under the square root to be non-negative. This leads to the inequality: \[ -8 - 8x \geq 0 \] Solving this, we get: \[ -8 \geq 8x \quad \Rightarrow \quad -1 \geq x \quad \Rightarrow \quad x \leq -1 \] Thus, the domain of \( f(x) \) is \( (-\infty, -1] \).
