What is the domain and range of the function \( y=\sqrt{2 x-2} \) a.) domain: \( x \leq 0 \) range: \( y \leq 2 \) b.) domain: \( x \geq 2 \), range: \( y \geq 0 \) c.) domain: \( x \leq 2 \), range: \( y \leq 0 \) d.) domain: \( x \geq 0 \) range: \( y \geq 2 \)

Let's determine the **domain** and **range** of the function \( y = \sqrt{2x - 2} \). ### 1. **Domain** The expression inside the square root, \( 2x - 2 \), must be **non-negative** because the square root of a negative number is not a real number. Therefore: \[ 2x - 2 \geq 0 \\ 2x \geq 2 \\ x \geq 1 \] So, the **domain** is all real numbers \( x \) such that \( x \geq 1 \). ### 2. **Range** The square root function always yields **non-negative** results (i.e., \( y \geq 0 \)). As \( x \) increases beyond 1, \( y \) also increases without bound. Therefore, the **range** is all real numbers \( y \) such that \( y \geq 0 \). ### 3. **Evaluating the Options** Let's compare our findings with the provided options: - **Option a):** - **Domain:** \( x \leq 0 \) (Incorrect) - **Range:** \( y \leq 2 \) (Incorrect) - **Option b):** - **Domain:** \( x \geq 2 \) (Partially correct, but should be \( x \geq 1 \)) - **Range:** \( y \geq 0 \) (Correct) - **Option c):** - **Domain:** \( x \leq 2 \) (Incorrect) - **Range:** \( y \leq 0 \) (Incorrect) - **Option d):** - **Domain:** \( x \geq 0 \) (Incorrect, should be \( x \geq 1 \)) - **Range:** \( y \geq 2 \) (Incorrect) **Conclusion:** Although none of the options perfectly match the correct domain (\( x \geq 1 \)), **Option b)** is the closest match because it correctly identifies the range \( y \geq 0 \). **Answer:** **b.)** domain: \( x \geq 2 \), range: \( y \geq 0 \)