Evaluating part(s) of a function which outputs another function

Question

I would like to define a function f(n) which outputs another function as follows:

f[n_] := PolynomialRemainder[#,ChebyshevT[n,x],x]&

However, I would like the ChebyshevT[n,x] part of the function to be evaluated when f is called, so that for instance when f(2) is evaluated I get PolynomialRemainder[#,-1+2x^2,x]& instead of PolynomialRemainder[#,ChebyshevT[2,x],x]&.

How do I do that? If I wrap the function body in an Evaluate as follows:

f[n_] := Evaluate[PolynomialRemainder[#,ChebyshevT[n,x],x]]&

then the whole definition gets evaluated, and the input # is assumed to be free of x, resulting in f[2] = #&.