That the function has an inverse requires that there be one and only one solution \(\displaystyle x\) that gives the value \(\displaystyle g(y)\), and then \(\displaystyle g(x)=g(y). \) As g(x) is a quadratic this is the case as long as \(\displaystyle x<x_0\), where \(\displaystyle (x_0,g(x_0))\) is the lowest point on the graph of \(\displaystyle y=g(x).\)