a good way to check is the "horizontal line test" ...
if a horizontal line can be drawn anywhere and only crosses the function once, then that function has an inverse.
most often, even functions cannot pass this test unless their domain is restricted. odd functions usually fare better, but not all odd functions can pass this test.
bottom line ... a function has to be "one to one" (each function y-value is generated by only one x-value) to be invertible.