What are the domain and range of the function? \[ y=|x-3| \] \begin{tabular}{|c|} Domain: \( \{x \mid x \in \mathbb{R}\} \); Range: \( \{y \mid y \geq 0\} \) \\ Domain: \( \{x \mid x \geq 3\} \); Range: \( \{y \mid y>0\} \) \\ Domain: \( \{x \mid x \in \mathbb{R}\} \); Range: \( \{y \mid y>0\} \) \\ Domain: \( \{x \mid x>3\} \); Range: \( \{y \mid y \geq 0\} \) \\ \\ \hline\end{tabular}

To determine the **domain** and **range** of the function \( y = |x - 3| \), let's analyze each part step-by-step. ### **Domain:** The domain of a function consists of all possible input values (values of \( x \)) for which the function is defined. - The absolute value function \( |x - 3| \) is defined for **all real numbers**. There are no restrictions on \( x \) because you can subtract 3 from any real number and then take its absolute value. **Thus, the domain is:** \[ \{ x \mid x \in \mathbb{R} \} \] ### **Range:** The range of a function consists of all possible output values (values of \( y \)) that the function can produce. - The absolute value function always yields a **non-negative** result because it measures the distance from zero. - The smallest value \( y \) can take is **0** (when \( x = 3 \)), and it can extend to **infinity** as \( x \) moves away from 3 in either direction. **Thus, the range is:** \[ \{ y \mid y \geq 0 \} \] ### **Conclusion:** Among the given options, the first one accurately represents both the domain and range of the function \( y = |x - 3| \). \[ \boxed{\text{Domain: } \{x \mid x \in \mathbb{R}\};\ \text{Range: } \{y \mid y \geq 0\}} \]