Let $ f,g : A \rightarrow A (A \subset R)$ be two bijections. Give examples that f+g and fg are not bijections

Question

Let $ f,g : A \rightarrow A (A \subset R)$ be two bijections. Give examples that f+g and fg are not bijections.

I am not sure how to approach this question. I try to think of examples, but I can't think of any. Thanks for any help!

Answer 1

Hint: Try $A=[-1,1]$, $f(x)=x, g(x)=-x$.

Then $f(x)+g(x) = 0$ is not bijection and $f(x) g(x)=-x^2$ also is not bijection since $f(1) g(1)=f(-1) g(-1).$

Answer 2

Just wish to contribute that in general for these problems it is very convenient to think of graphs on the Cartesian Plane.

Graphs that do not pass the vertical line test (no two y values for one x value) are not functions.

and

Graphs that do not pass the horizontal line test (similarly, no two x values for one y value) are not injective.

Then it becomes a little easier to think of functions that are not bijective. Consider the graph of x^2. This will not pass the horizontal line test, and therefore any function that maps to it cannot map back (as noted by Leox above).