One-to-One Question

Question

For each statement below, explain why it must be True, or provide a counterexample to show that it can be False.

(Note: 𝑓 is a function from  ℝ to  ℝ .)

1. If  𝑓 is even then  𝑓 is not one-to-one.
2. If  𝑓 is odd then  𝑓  is one-to-one.

Answer 1

1) The statement is correct since by definition an even function satisfies f(x)=f(-x) and so f is not 1-1.

2) The statement is false since the function f(x)=0 for all x is an odd function that is not 1-1. For a more interesting counterexample look at the odd function f(x)=x^{3}-x.