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## Homework Statement

Provide a counter example to the false assertion:

Suppose that f and g are functions and f ◦ g is invertible. Then f and g are invertible.

## Homework Equations

Definitions:

An invertible function is 1-1 and onto

If the image of g is not contained in the domain of f then f ◦ g is not a legal expression.

## The Attempt at a Solution

Let f(x)=x^2 over R and g(x)=sqrt(x) over x>0|R, then f(g(x))=x which is 1-1 and onto over x>0|R and is therefore invertible.

f(x) is not 1-1 over R thus it is not invertible providing the counter example to f and g are invertible.

My question: The function I defined as f is not invertible but the composition works over a subset of f's domain due to g. Does this mean that f's invertibility should only be considered over the reduced domain too?

I suspect not because I provided that exact wording of the problem and it says I only need to provide a function definition which is not invertible on its own but which is invertible in a composition.

I have a strong hunch I am over thinking this...