To determine the range of the function [tex]\( f(x) = -|x-4| + 5 \)[/tex], we need to analyze how the expression behaves as [tex]\( x \)[/tex] varies over all possible real numbers.
1. Understanding the Absolute Value Function:
First, we consider the absolute value function [tex]\( |x-4| \)[/tex]. The absolute value function [tex]\( |x-4| \)[/tex] measures the distance between [tex]\( x \)[/tex] and 4 on the real number line. This distance is always non-negative, meaning [tex]\( |x-4| \geq 0 \)[/tex].
2. Effect of the Absolute Value on the Expression:
The function can be rewritten as [tex]\( f(x) = -|x-4| + 5 \)[/tex]. The term [tex]\( |x-4| \)[/tex] can take any value from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex]. As [tex]\( x \)[/tex] gets closer to 4, [tex]\( |x-4| \)[/tex] approaches 0. Conversely, as [tex]\( x \)[/tex] moves away from 4, [tex]\( |x-4| \)[/tex] becomes larger.
3. Analyze Maximum Value:
We need to find when [tex]\( f(x) \)[/tex] achieves its maximum value. Notice that:
[tex]\[
f(4) = -|4-4| + 5 = -0 + 5 = 5
\][/tex]
Therefore, the maximum value of [tex]\( f(x) \)[/tex] is 5, which occurs when [tex]\( x = 4 \)[/tex].
4. Analyze Minimum Behavior:
As [tex]\( x \)[/tex] moves further from 4, [tex]\( |x-4| \)[/tex] increases, which causes [tex]\( -|x-4| \)[/tex] to become more negative. Thus, [tex]\( -|x-4| + 5 \)[/tex] becomes smaller and smaller (more negative) as [tex]\( x \)[/tex] moves away from 4 in either direction (towards [tex]\(\infty\)[/tex] or [tex]\(-\infty\)[/tex]).
5. Conclusion on the Range:
- Since [tex]\( -|x-4| \)[/tex] can decrease without bound (towards [tex]\(-\infty\)[/tex]), [tex]\( -|x-4| + 5 \)[/tex] can go as low as [tex]\(-\infty\)[/tex].
- The function value 5 is included in the range because it is specifically achievable when [tex]\( x = 4 \)[/tex].
Putting it all together, the range of the function [tex]\( f(x) = -|x-4| + 5 \)[/tex] is the set of all real numbers less than or equal to 5.
Therefore, the correct answer is:
[tex]\[
\boxed{(-\infty, 5]}
\][/tex]
