The range of \( f(x)=|x| \) is \( y \geq 0 \). If \( a \leq 0 \) for \( g(x)=a|x| \), what is the range of function \( g \) ? A. \( y \leq a \)

For the function \( g(x) = a|x| \) where \( a \leq 0 \), let's determine its range. 1. **Understanding \( |x| \):** The absolute value function \( |x| \) always yields non-negative values, i.e., \( |x| \geq 0 \). 2. **Scaling by \( a \):** When you multiply \( |x| \) by \( a \) (where \( a \leq 0 \)), the output is scaled by a non-positive factor. This means: \[ g(x) = a|x| \leq 0 \] - If \( a < 0 \), as \( |x| \) increases, \( g(x) \) decreases without bound (approaches negative infinity). - If \( a = 0 \), \( g(x) \) is always 0. 3. **Range of \( g(x) \):** Combining the above points: - The maximum value of \( g(x) \) is 0 (when \( |x| = 0 \)). - There is no lower bound if \( a < 0 \), as \( g(x) \) can be as negative as desired. Therefore, the range of \( g(x) \) is: \[ y \leq 0 \] **Conclusion:** If \( a \leq 0 \) for \( g(x) = a|x| \), the range of \( g \) is \( y \leq 0 \). Answer: Its range consists of all real numbers y with y ≤ 0.