Solutions

DL-C04-C-001

ReLU maps:

[-3, 2, 0, 5] -> [0, 2, 0, 5]

The positive outputs are 2 and 5, so there are 2.

DL-C04-C-002

relu(-11) = max(0, -11) = 0

DL-C04-C-003

Sigmoid maps real inputs to values between 0 and 1, so it is bounded.

DL-C04-C-004

First compute the pre-activation:

x . w + b = 2 x 1 + 2 x (-3) + 5
          = 2 - 6 + 5
          = 1

Then apply ReLU:

relu(1) = 1

DL-C04-C-005

y = -2h
  = -2(4x)
  = -8x

The multiple is -8.

DL-C04-C-006

Stacked linear layers without activations collapse into one linear map. Nonlinear activations bend the computation between layers, so depth can build richer functions.