What are the domain and range of \( f(x)=|x-3|+6 \) ? Domain: \( \{x \mid x \) is all real numbers \( \} \) Range: \( \{y \mid y \geq 6\} \) Domain: \( \{x \mid x \geq 3\} \) Range: \( \{y \mid y \geq 6\} \) Domain: \( \{x \mid x \) is all real numbers \( \} \) Range: \( \{y \mid y \geq-6\} \) Domain: \( \{x \mid x \geq 3\} \) Range: \( \{y \mid y \geq-6\} \)

The function in question is \( f(x) = |x - 3| + 6 \). Let's analyze its **domain** and **range**. ### **Domain** The domain of a function consists of all possible input values (x-values) for which the function is defined. - The absolute value function \( |x - 3| \) is defined for all real numbers \( x \). - Adding 6 does not restrict the domain further. **Therefore, the domain is:** \[ \{x \mid x \text{ is all real numbers}\} \] ### **Range** The range of a function consists of all possible output values (y-values). - The expression \( |x - 3| \) is always **non-negative**, meaning \( |x - 3| \geq 0 \). - When you add 6 to \( |x - 3| \), the smallest possible value of \( f(x) \) becomes \( 0 + 6 = 6 \). - As \( |x - 3| \) increases, \( f(x) \) increases without bound. **Therefore, the range is:** \[ \{y \mid y \geq 6\} \] ### **Conclusion** Among the provided options, the correct one is: - **Domain:** \( \{x \mid x \text{ is all real numbers}\} \) - **Range:** \( \{y \mid y \geq 6\} \) This corresponds to the **first option** you listed.