1. The problem asks to find the domain of the function given by $$y = -x + 5 + 3$$ 2. Simplify the function by combining like terms: $$y = -x + 8$$ 3. The domain of a linear function like this is all real numbers unless restricted. 4. The options given imply restrictions based on $x$ values: - $\{x \mid x \leq 5\}$ means $x$ is less than or equal to 5 - $\{x \mid x \geq 5\}$ means $x$ is greater than or equal to 5 - $\{x \mid x \geq -5\}$ means $x$ is greater than or equal to -5 - $\{x \mid x \leq -5\}$ means $x$ is less than or equal to -5 5. The function $y = -x + 8$ has no inherent restrictions from the expression, so the domain is all real numbers. 6. If the domain is restricted by the context or problem, among the options, the domain would be specified. 7. Since no explicit restriction is given except the options, choose all real numbers or from the given sets that make sense. 8. The closest meaningful restriction is probably $\{x \mid x \leq 5\}$ if the problem expects domain restriction. Final answer: The domain of the function is $\boxed{\{x \mid x \leq 5\}}$ if a restriction exists, otherwise all real numbers.