there can only be an inverse if each x value corresponds to a single y value.

Your graph has a turning point a . starting from x=0:

as you increase x, y increases

after , y starts to decrease again

Can you see that this means there must be some y values that occur twice, once to the right of and once to the left?

it might be clearer on this graph:

y=3x/(5+x^2) from -10,10 - Wolfram|Alpha