1. **Problem:** Find the domain and range of the function $f(x) = 2 - |x - 5|$. 2. **Formula and rules:** The domain of a function is the set of all possible input values ($x$) for which the function is defined. The range is the set of all possible output values ($f(x)$). 3. **Domain:** Since $|x - 5|$ is defined for all real numbers, the domain of $f$ is all real numbers, $\mathbb{R}$. 4. **Range:** The absolute value $|x - 5|$ is always non-negative, so $|x - 5| \geq 0$. Therefore, $f(x) = 2 - |x - 5| \leq 2$ because subtracting a non-negative number from 2 can only decrease or keep the value at 2. 5. The smallest value of $f(x)$ occurs when $|x - 5|$ is very large, tending to infinity, so $f(x)$ tends to $-\infty$. 6. Hence, the range is $(-\infty, 2]$. 7. **Answer:** Domain = $\mathbb{R}$, Range = $(-\infty, 2]$ which corresponds to option b).