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

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 .)

(a) If 𝑓 is even then 𝑓 is not one-to-one.

(b) If 𝑓 is odd then 𝑓 is one-to-one.

Answer 1

(a) If 𝑓 is even then 𝑓 is not one-to-one. This must be true.

The statement is true for all even functions.  𝑓 is even means that 𝑓(x) = 𝑓(–x) for all x.

A function 𝑓(x) that has same value for two different values of x is not one-to-one.

(b) If 𝑓 is odd then 𝑓 is one-to-one. This can be false.

The statement is true for many odd functions.

For example, it is true for 𝑓(x) = axⁿ for any real number a and any odd integer n.

However, there are some odd function that are not one-to-one. For example the trigonometric functions sin x, tan x, and csc x are all odd, but they are not one-to-one.