Real Analysis – What Does It Mean to Extend a Function?

Question

What does it mean to extend a function? Can someone please give an example?

Thanks in advance!

Answer 1

A good example might be $\sqrt{x}$. We cannot put a negative $x$ in a square root if we only care about real numbers. But, we can define a function $\sqrt{|x|}$. If $x\geq 0$ this is the same function as before since, in this case, $|x|=x$. But, the function can now take in negative numbers.