One to One Function

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) If f is even, then f(-1) = f(1), which means f cannot be one-to-one. So the statement is TRUE.

2) If f is odd, f is not necessarily one-to-one. For example, the function f(x) = 0 is odd, but not one-to-one. So the statement is FALSE.