Question

Is it true that if g is one-to-one then f o g must be one-to-one? Give a proof or a counterexample.

Answer 1

It is not true in general that if `g` is one-to-one then `f \circ g` must be one-to-one.

Explanation:

This is not true in general.

Notice that the identity function `g(x)=x` is a one-to-one function. If `f(x)` is any function that is not one-to-one, `f(g(x)) = f(x)` and is therefore also not one-to-one.