Question

suppose that the functions u and w are defined as follows.
u(x) = x² + 6
w(x) = √(x + 1)
find the following.
(u ∘ w)(3) =
(w ∘ u)(3) =

Explanation:

Step1: Define function composition

$(u \circ w)(x) = u(w(x))$, $(w \circ u)(x) = w(u(x))$

Step2: Compute $(u \circ w)(3)$ first

First find $w(3)$:
$w(3) = \sqrt{3+1} = \sqrt{4} = 2$
Substitute into $u(x)$:
$u(w(3)) = u(2) = 2^2 + 6 = 4 + 6 = 10$

Step3: Compute $(w \circ u)(3)$

First find $u(3)$:
$u(3) = 3^2 + 6 = 9 + 6 = 15$
Substitute into $w(x)$:
$w(u(3)) = w(15) = \sqrt{15+1} = \sqrt{16} = 4$

Answer:

$(u \circ w)(3) = 10$
$(w \circ u)(3) = 4$