Let's break down the problem step by step: ### Given Function \[ f(x) = 2|x-1| \] We are asked to evaluate: 1. \( f(|x+1|) \) 2. \( f(|x^2 + 2|) \) --- ## 1. Evaluate \( f(|x+1|) \): Substitute \( |x+1| \) into \( f(x) \): \[ f(|x+1|) = 2 \left| |x+1| - 1 \right| \] Now, simplify inside the absolute value: \[ = 2 \left| |x+1| - 1 \right| \] **Final answer:** \[ \boxed{f(|x+1|) = 2 \left| |x+1| - 1 \right|} \] --- ## 2. Evaluate \( f(|x^2 + 2|) \): Substitute \( |x^2 + 2| \) into \( f(x) \): \[ f(|x^2 + 2|) = 2 \left| |x^2 + 2| - 1 \right| \] **Final answer:** \[ \boxed{f(|x^2 + 2|) = 2 \left| |x^2 + 2| - 1 \right|} \] --- ## **Summary of Answers** \[ \boxed{ \begin{aligned} &f(|x+1|) = 2 \left| |x+1| - 1 \right| \\ &f(|x^2 + 2|) = 2 \left| |x^2 + 2| - 1 \right| \end{aligned} } \] These are the step-by-step evaluations of the function at the given points. Let me know if you'd like to see these evaluated for a specific value of \(x\)!