Question
Which value is in the domain of \( f(x) ? \)
\( f(x)=\left\{\begin{array}{ll}2 x+5, & -6
Ask by Wells Carrillo. in the United States
Jan 24,2025
Ask by Wells Carrillo. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
4 is in the domain of \( f(x) \).
Solution
To determine which value is in the domain of the function \( f(x) \), let's analyze the given piecewise function:
\[
f(x) = \begin{cases}
2x + 5, & -6 < x \leq 0 \\
-2x + 3, & 0 < x \leq 4
\end{cases}
\]
### Step-by-Step Analysis:
1. **Domain of Each Piece:**
- **First Piece:** \( 2x + 5 \) is defined for \( -6 < x \leq 0 \).
- **Second Piece:** \( -2x + 3 \) is defined for \( 0 < x \leq 4 \).
2. **Combined Domain:**
- The overall domain of \( f(x) \) is \( -6 < x \leq 4 \).
- **Note:** The value \( x = 0 \) is included in the first piece but not in the second.
3. **Evaluating the Given Options:**
- **-7:** \( -7 \) is **not** in the domain since \( -7 < -6 \).
- **-6:** \( -6 \) is **not** in the domain because the inequality is strict (\( -6 < x \)).
- **4:** \( 4 \) **is** in the domain as it satisfies \( -6 < 4 \leq 4 \).
- **5:** \( 5 \) is **not** in the domain since \( 5 > 4 \).
### Conclusion:
Among the provided options, **4** is the only value that lies within the domain of \( f(x) \).
**Answer:** 4
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
Let’s get into the fun world of piecewise functions! The function \( f(x) \) is defined in two segments: for values between -6 and 0 (exclusive of -6 but inclusive of 0) and for values between 0 and 4 (exclusive of 0 but inclusive of 4). So, any inputs that fall into -6 < x ≤ 0 or 0 < x ≤ 4 are perfectly acceptable! Now, checking the provided options: -7 is too low, -6 is not included, 5 is out of bounds. However, 4 sits comfortably at the top of the second segment. So, the value 4 is indeed in the domain of \( f(x) \).
