MAST10005: Prove That F is Injective You May Assume that the Function g: Calculus 1 Assignment, UoM, Singapore

Completing Question 1 before proceeding will make Question 3 easier because you will have already completed relevant examples in Problem 3 of the Web Work Part.

2. Consider the function f : R −→ R 2 defined by f(x) = (10 sin(x), x3)

(a) Prove that f is injective. You may assume that the function g: R −→ R defined by g(x) = x 3 is injective, but you should state clearly where you use this fact. [Hint: For (a, b),(c, d) ∈ R 2 , when is it true that (a, b) = (c, d)?]

3. Consider the function f: R 2 −→ R defined by f(s, t) = s 2 − t 2

(a) Prove that f is surjective.

(b) Is f injective? Explain your reasoning.

4. Define f : C −→ R by f(z) = I’m(z) and g : R −→ C by g(x) = ix.

(a) Find a formula for f ◦ g.

(b) Find a formula for g(f(z)) by expressing z ∈ C as z = x + iy. Hence give a simple description of the set

A = {z ∈ C : g(f(z)) = z}.

(c) Is g the inverse of f? Explain your answer.