Let the function f(x) be defined as f(x) = array ccc x^2 - 4 & if & x If g(x) = f(|x|) - |f(x)| , then the range of the function g(x) on the interval [-3, 5] is

Step-by-step solution

For x [-3, 5] , since |x| 0 , we have f(|x|) = |x| - 4 . Now, we evaluate |f(x)| by considering the sign of f(x) in different sub-intervals: 1) For x [-3, -2] : f(x) = x^2 - 4 0 , so |f(x)| = x^2 - 4 . 2) For x (-2, 0) : f(x) = x^2 - 4 3) For x [0, 4] : f(x) = x - 4 0 , so |f(x)| = 4 - x . 4) For x (4, 5] : f(x) = x - 4 > 0 , so |f(x)| = x - 4 . Next, we find g(x) = f(|x|) - |f(x)| in each sub-interval: 1) x [-3, -2] : g(x) = (-x - 4) - (x^2 - 4) = -x^2 - x . The derivative is -2x - 1 > 0 , so g(x) is increasing. T