Let's analyze each of the given functions to determine their ranges.
1. For the function [tex]\( f(x) = (x-4)^2 + 5 \)[/tex]:
- This is a quadratic function that opens upwards (since the coefficient of [tex]\((x-4)^2\)[/tex] is positive).
- The vertex of this parabola is at [tex]\((4, 5)\)[/tex].
- Since it opens upwards, the minimum value of the function is [tex]\(5\)[/tex].
- Thus, the range of this function is [tex]\(\{y \mid y \ge 5\}\)[/tex].
2. For the function [tex]\( f(x) = -(x-4)^2 + 5 \)[/tex]:
- This is a quadratic function that opens downwards (since the coefficient of [tex]\((x-4)^2\)[/tex] is negative).
- The vertex of this parabola is at [tex]\((4, 5)\)[/tex].
- Since it opens downwards, the maximum value of the function is [tex]\(5\)[/tex].
- Thus, the range of this function is [tex]\(\{y \mid y \le 5\}\)[/tex].
3. For the function [tex]\( f(x) = (x-5)^2 + 4 \)[/tex]:
- This is a quadratic function that opens upwards (since the coefficient of [tex]\((x-5)^2\)[/tex] is positive).
- The vertex of this parabola is at [tex]\((5, 4)\)[/tex].
- Since it opens upwards, the minimum value of the function is [tex]\(4\)[/tex].
- Thus, the range of this function is [tex]\(\{y \mid y \ge 4\}\)[/tex].
4. For the function [tex]\( f(x) = -(x-5)^2 + 4 \)[/tex]:
- This is a quadratic function that opens downwards (since the coefficient of [tex]\((x-5)^2\)[/tex] is negative).
- The vertex of this parabola is at [tex]\((5, 4)\)[/tex].
- Since it opens downwards, the maximum value of the function is [tex]\(4\)[/tex].
- Thus, the range of this function is [tex]\(\{y \mid y \le 4\}\)[/tex].
After analyzing these functions, we can see that the function [tex]\( f(x) = -(x-4)^2 + 5 \)[/tex] has a range of [tex]\(\{y \mid y \le 5\}\)[/tex].
Therefore, the function that has the range [tex]\(\{y \mid y \leq 5\}\)[/tex] is [tex]\(\boxed{2}\)[/tex].
