What are the domain and range of the function defined by \( y=|x+3|-2 \) notation and interval notation? \( \begin{array}{ll}\text { a } \\ \text { domain: }\{x \mid x \geq-3\} ;[-3, \infty) \\ \text { range: }\{y \mid y \geq 0\} ;[0, \infty) \\ \text { domain: }\{x \mid x \geq-3\} ;[-3, \infty) \\ & \text { range: }\{y \mid y \geq 2\} ;[2, \infty) \\ \text { o domain: }\{x \mid x \text { is a real number }\} ;(-\infty, \infty) \\ & \text { range: }\{y \mid y \text { is a real number }\} ;(-\infty, \infty) \\ \text { d domain: }\{x \mid x \text { is a real numbert }\} ;(-\infty, \infty)\end{array} \) range: \( \{y \mid y \geq-2\} ;[-2, \infty) \)

To determine the domain and range of the function \( y = |x + 3| - 2 \), we can analyze the function step by step. ### Domain: The expression \( |x + 3| \) is defined for all real numbers \( x \). Therefore, the domain of the function is all real numbers. - **Domain in set notation**: \( \{x \mid x \text{ is a real number}\} \) - **Domain in interval notation**: \( (-\infty, \infty) \) ### Range: Next, we analyze the range of the function. The absolute value function \( |x + 3| \) has a minimum value of 0, which occurs when \( x + 3 = 0 \) (i.e., \( x = -3 \)). At \( x = -3 \): \[ y = |(-3) + 3| - 2 = |0| - 2 = -2 \] As \( x \) moves away from -3 in either direction, \( |x + 3| \) increases, and thus \( y \) will also increase. Therefore, the minimum value of \( y \) is -2, and as \( x \) approaches positive or negative infinity, \( y \) approaches positive infinity. - **Range in set notation**: \( \{y \mid y \geq -2\} \) - **Range in interval notation**: \( [-2, \infty) \) ### Summary: - **Domain**: - Set notation: \( \{x \mid x \text{ is a real number}\} \) - Interval notation: \( (-\infty, \infty) \) - **Range**: - Set notation: \( \{y \mid y \geq -2\} \) - Interval notation: \( [-2, \infty) \)