Question

Suppose that the function u and w are defined as follows.

Answer 1

That funky circle in the middle is the composition of the function. It asks you to take a function as an input and to yield an output that's another function. It's one of the five function operations, along with adding, subtracting, multiplying, and dividing.

When you compose, you might find the notation w(u(x)) easier to understand. It's saying evaluate u then evaluate w.

For our functions, the compositions are:

u(w(x)) = u(2x²) = -(2x²) - 2 = -2x² - 2

w(u(x)) = w(-x - 2) = 2(-x - 2)² = 2(x² + 4x + 4) = =2x²+ 8x +8

Now we evaluate each composition at 4.

u(w(4)) = -2(4²) - 2 = -2(16) - 2 = -32 -2 = -34

w(u(4)) = -2(2²) +8(2) + 8 = -2(4) + 16 + 8 = -8 + 16 + 8 = 16.

Thus, u(w(4)) = -34 and w(u(4)) = 16.