y=2x2+x-3=(2x+3)(x-1). So if x=1 or x=-3/2, y=0. The inverse function would need to map 0 to different values, that is, one input maps to more than one output. But all functions must map a particular input to one and only one output, so there can be no inverse function. In this case there would be two possible inverses. Let's see what happens.
2x2+x-3-y=0, x=(-1±√(1+8(y+3))/4=(-1±√(y+25))/4.
Let g(y)=(√(8y+25)-1)/4, h(y)=-(√(8y+25)+1)/4, making the inverse functions:
g(x)=(√(8x+25)-1)/4, h(y)=-(√(8x+25)+1)/4. Neither function would be sufficient on its own as an inverse. Each of these functions maps to half of the original curve. The only unique point is where the two halves join at (-¼,-25/8), the minimum point, when we have the square root of zero.