Let g ∈ G, where G is a group. Prove that the function f : G → G given by f (h) = ghg−1 isan isomorphism from G to itself (i.e. an automorphism of G).

Let f(x) and g (x) be one-to-one functions, and let h (x) = g ( f(x)). Show that the inverse functionof h (x) is h^−1 (x) = f^ −1(g^−1 (x))

For general vector fields F and G prove that∇ × (F + G) = ∇ × F + ∇ × G,[6 marks]∇ • (F × G) = G • (∇ × F) − F • (∇ × G)

Let f, g : R → R be given functions. Suppose that f and g are continuous at c ∈ R.Prove that the functionl(x) := inf{f (x), g(x)}, x ∈ R,is continuous at c.

Evaluate the difference quotient for the given function. Simplify your answer.f(x) = −x3,    f(a + h) − f(a)h

Let f(x) be a real value function such that f(x)=2x−1x−2∀ x ∈ (2,∞) and g(x)=x2+1x+(f(x))2+1f(x)∀ x>2, then minimum value of g(x) is