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]
