Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine the range of the function [tex]\( f(x) = -|x-4| + 5 \)[/tex], let's analyze it step by step.
1. Understand the absolute value function:
The function [tex]\( f(x) = -|x-4| + 5 \)[/tex] contains an absolute value component [tex]\(|x-4|\)[/tex]. Absolute values are always non-negative, meaning [tex]\(|x-4| \geq 0\)[/tex] for any real number [tex]\(x\)[/tex].
2. Effect of the negative sign:
When we apply the negative sign, [tex]\(-|x-4|\)[/tex], the range of [tex]\(|x-4|\)[/tex] changes accordingly:
- Since [tex]\(|x-4|\)[/tex] is always non-negative ([tex]\(|x-4| \geq 0\)[/tex]), [tex]\(-|x-4|\)[/tex] will be non-positive ([tex]\(-|x-4| \leq 0\)[/tex]), meaning [tex]\(-|x-4|\)[/tex] takes values from [tex]\(0\)[/tex] to negative infinity.
- In other words, [tex]\(-|x-4|\)[/tex] can take any value from [tex]\(-\infty\)[/tex] up to [tex]\(0\)[/tex] (inclusive).
3. Shifting the range by adding 5:
Adding 5 to [tex]\(-|x-4|\)[/tex] will shift its entire range upwards by 5 units:
- If [tex]\(-|x-4|\)[/tex] ranges from [tex]\(-\infty\)[/tex] to [tex]\(0\)[/tex], then [tex]\(-|x-4| + 5\)[/tex] will shift this range to [tex]\(-\infty + 5\)[/tex] to [tex]\(0 + 5\)[/tex].
- Therefore, after adding 5, the new range becomes [tex]\(-\infty\)[/tex] to [tex]\(5\)[/tex] (inclusive).
4. Conclusion:
The highest value of [tex]\( f(x) \)[/tex] occurs when [tex]\(|x-4| = 0\)[/tex], which results in [tex]\( f(x) = 5 \)[/tex]. Thus, the value 5 is included in the range.
Given this analysis, we conclude that the range of the function [tex]\( f(x) = -|x-4| + 5 \)[/tex] is [tex]\((-\infty, 5]\)[/tex].
Hence, the correct option is:
A. [tex]\( (-\infty, 5] \)[/tex]
1. Understand the absolute value function:
The function [tex]\( f(x) = -|x-4| + 5 \)[/tex] contains an absolute value component [tex]\(|x-4|\)[/tex]. Absolute values are always non-negative, meaning [tex]\(|x-4| \geq 0\)[/tex] for any real number [tex]\(x\)[/tex].
2. Effect of the negative sign:
When we apply the negative sign, [tex]\(-|x-4|\)[/tex], the range of [tex]\(|x-4|\)[/tex] changes accordingly:
- Since [tex]\(|x-4|\)[/tex] is always non-negative ([tex]\(|x-4| \geq 0\)[/tex]), [tex]\(-|x-4|\)[/tex] will be non-positive ([tex]\(-|x-4| \leq 0\)[/tex]), meaning [tex]\(-|x-4|\)[/tex] takes values from [tex]\(0\)[/tex] to negative infinity.
- In other words, [tex]\(-|x-4|\)[/tex] can take any value from [tex]\(-\infty\)[/tex] up to [tex]\(0\)[/tex] (inclusive).
3. Shifting the range by adding 5:
Adding 5 to [tex]\(-|x-4|\)[/tex] will shift its entire range upwards by 5 units:
- If [tex]\(-|x-4|\)[/tex] ranges from [tex]\(-\infty\)[/tex] to [tex]\(0\)[/tex], then [tex]\(-|x-4| + 5\)[/tex] will shift this range to [tex]\(-\infty + 5\)[/tex] to [tex]\(0 + 5\)[/tex].
- Therefore, after adding 5, the new range becomes [tex]\(-\infty\)[/tex] to [tex]\(5\)[/tex] (inclusive).
4. Conclusion:
The highest value of [tex]\( f(x) \)[/tex] occurs when [tex]\(|x-4| = 0\)[/tex], which results in [tex]\( f(x) = 5 \)[/tex]. Thus, the value 5 is included in the range.
Given this analysis, we conclude that the range of the function [tex]\( f(x) = -|x-4| + 5 \)[/tex] is [tex]\((-\infty, 5]\)[/tex].
Hence, the correct option is:
A. [tex]\( (-\infty, 5] \)[/tex]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.