Suppose that $p(x)=1/4x^4−2/3x^3-5/2x^2+6x-1/12 $withDom(p)=[1,2].Define$q(x)=p^−1(x)$. Show, algebraically, why q(x) exists

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I don't know where to start. What does it means to define $q(x) = p^-1(x)$?

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A function is invertible when it is 1-1 on it's range. So simply check that $p(x)$ is 1-1, meaning that $p(a) = p(b)$ implies $a = b$, and you're done.