Step-by-step explanation:
(a)
g(x) = d/dx [2/√π ∫₀ˣ e^(-t²) dt]
g(x) = 2/√π e^(-x²)
g(-x) = 2/√π e^(-(-x)²) = 2/√π e^(-x²)
g(-x) = g(x)
Therefore, g(x) is even.
(b)
erf(x) = 2/√π ∫₀ˣ e^(-t²) dt
erf(x) = ∫₀ˣ g(t) dt
erf(-x) = ∫₀⁻ˣ g(t) dt
erf(-x) = -∫₋ₓ⁰ g(t) dt
Since g(t) is even, it is symmetrical about x=0.
erf(-x) = -∫₀ˣ g(t) dt
erf(-x) = -erf(x)
