Gaussian Quadrature Points

Question

I am trying to determine the transformation necessary to map a given set of Gauss points on the interval $[a,b]$ to the corresponding Gaussian quadrature points on the interval $[α,β]$. Here is what I have done so far. $$ φ:[a,b]\to [α,β]\text{ which is defined by }φ(x)=(((x-a)(β-α))/(b-a))+α. $$ This defines $φ(x)$ as a one to one, or bijection between the two intervals provided and is strictly increasing. It can be noted that $φ(a)=α$ and $φ(b)=β$. Is this enough to prove that points in $[a,b]$ maps to $[α,β]$, or do I need a stronger argument?

Answer 1

Yes, that is enough. Simply note that $φ(a)=\alpha$ and $φ(b)=\beta$, as you mentioned, and that your function is monotone for $a<b$ and $\alpha<\beta$.

A bit redundant, but it is a line and as you say it is a bijection, so that is indeed enough.