Question

u(x) = 2x + 1
w(x) = -x² - 2
find the following.
your answer
(u ∘ w)(-5) =
(w ∘ u)(-5) =

Explanation:

Part 1: Find \((u \circ w)(-5)\)

Step 1: Recall the definition of composition of functions

The composition \((u \circ w)(x)\) means \(u(w(x))\). So first, we need to find \(w(-5)\), and then substitute that result into \(u(x)\).

Step 2: Calculate \(w(-5)\)

Given \(w(x) = -x^2 - 2\), substitute \(x = -5\):
\(w(-5)=-(-5)^2 - 2\)
First, calculate \((-5)^2 = 25\), so:
\(w(-5)=-25 - 2=-27\)

Step 3: Calculate \(u(w(-5)) = u(-27)\)

Given \(u(x) = 2x + 1\), substitute \(x = -27\):
\(u(-27)=2(-27)+1=-54 + 1=-53\)

Part 2: Find \((w \circ u)(-5)\)

Step 1: Recall the definition of composition of functions

The composition \((w \circ u)(x)\) means \(w(u(x))\). So first, we need to find \(u(-5)\), and then substitute that result into \(w(x)\).

Step 2: Calculate \(u(-5)\)

Given \(u(x) = 2x + 1\), substitute \(x = -5\):
\(u(-5)=2(-5)+1=-10 + 1=-9\)

Step 3: Calculate \(w(u(-5)) = w(-9)\)

Given \(w(x) = -x^2 - 2\), substitute \(x = -9\):
\(w(-9)=-(-9)^2 - 2\)
First, calculate \((-9)^2 = 81\), so:
\(w(-9)=-81 - 2=-83\)

Answer:

\((u \circ w)(-5)=\boxed{-53}\)
\((w \circ u)(-5)=\boxed{-83}\)