High School

Identify any x-values at which the absolute value function [tex]f(x)=4|x+2|[/tex] is not continuous:

x= _______

Identify any x-values at which the absolute value function [tex]f(x)=4|x+2|[/tex] is not differentiable:

x= _____

(Enter "none" if there are no x-values that apply; enter x-values as a comma-separated list, e.g., 1,3,5.)

Answer :

Final answer:

The function f(x) = 4|x+2| is continuous for all x-values, but it's not differentiable at x = -2.

Explanation:

The function f(x) = 4|x+2| is of the form |ax+b|, which is an absolute value function. Absolute value functions are always continuous for all real numbers, so for the given function, there are no x-values at which the function is not continuous.

However, the function is not differentiable at the point where x = -2. This is because the graph of absolute value function has a corner or cusp at x = -2, at which point the derivative does not exist. In a graphical representation, it is where the direction of the graph changes abruptly.

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