(d) To map \(y = f(x)\) to \(y = g(x)\):

1. First, apply a horizontal stretch by a scale factor of \(\frac{1}{2}\) because \(g(x) = f(2x)\).
2. Then, translate the graph vertically by 4 units upwards, as indicated by \(g(x) = f(2x) + 4\).

(e) To map \(y = f(x)\) to \(y = h(x)\):

1. First, translate the graph horizontally by \(-\frac{\pi}{2}\) units, as indicated by \(h(x) = 2f\left(x + \frac{1}{2}\pi\right)\).
2. Then, apply a vertical stretch by a scale factor of 2, as indicated by the multiplication by 2 in \(h(x) = 2f\left(x + \frac{1}{2}\pi\right)\).