Suppose that the functions \( u \) and \( w \) are defined as follows. \[ \begin{array}{l}u(x)=-2 x-2 \\ w(x)=-x^{2} \\ (w \circ u)(4)=\square \\ \text { Find the following. } \\ (u \circ w)(4)=\square\end{array} \]

To solve the problem, we need to evaluate the compositions of the functions \( u \) and \( w \) at the specified input. Let's break it down step by step. ### Step 1: Evaluate \( (w \circ u)(4) \) The composition \( (w \circ u)(x) \) means we first apply \( u \) to \( x \), and then apply \( w \) to the result of \( u(x) \). 1. **Calculate \( u(4) \)**: \[ u(x) = -2x - 2 \] Substituting \( x = 4 \): \[ u(4) = -2(4) - 2 = -8 - 2 = -10 \] 2. **Now calculate \( w(u(4)) = w(-10) \)**: \[ w(x) = -x^2 \] Substituting \( x = -10 \): \[ w(-10) = -(-10)^2 = -100 \] Thus, we have: \[ (w \circ u)(4) = -100 \] ### Step 2: Evaluate \( (u \circ w)(4) \) The composition \( (u \circ w)(x) \) means we first apply \( w \) to \( x \), and then apply \( u \) to the result of \( w(x) \). 1. **Calculate \( w(4) \)**: \[ w(4) = -4^2 = -16 \] 2. **Now calculate \( u(w(4)) = u(-16) \)**: \[ u(-16) = -2(-16) - 2 = 32 - 2 = 30 \] Thus, we have: \[ (u \circ w)(4) = 30 \] ### Final Answers - \( (w \circ u)(4) = -100 \) - \( (u \circ w)(4) = 30 \) So, the final results are: \[ (w \circ u)(4) = -100, \quad (u \circ w)(4) = 30 \]