The problem asks us to find the domain of the function $d(x) = 4 - \frac{1}{2}x$ for the range $(0, 4.5]$.

1. Problem Description

The problem asks us to find the domain of the function $d(x) = 4 - \frac{1}{2}x$ for the range $(0, 4.5]$.

2. Solution Steps

We are given the function $d(x) = 4 - \frac{1}{2}x$ and we want to find the domain for the range $(0, 4.5]$. This means we want to find the values of $x$ such that $0 < d(x) \leq 4.5$. We can set up the following inequality:

$0 < 4 - \frac{1}{2}x \leq 4.5$

First, consider the left inequality: $0 < 4 - \frac{1}{2}x$.

Subtracting 4 from both sides gives $-4 < -\frac{1}{2}x$.

Multiplying both sides by -2 and reversing the inequality sign gives $8 > x$, or $x < 8$.

Next, consider the right inequality: $4 - \frac{1}{2}x \leq 4.5$.

Subtracting 4 from both sides gives $-\frac{1}{2}x \leq 0.5$.

Multiplying both sides by -2 and reversing the inequality sign gives $x \geq -1$.

Combining these two inequalities, we have $-1 \leq x < 8$.

Therefore, the domain is $[-1, 8)$.

3. Final Answer

$[-1, 8)$