[INST] Let f(x)=2x+1. Find the sum of all x that satisfy the equation f^-1(x)=f(x^-1)./n [/INST]  ###Response: To find the sum of all x that satisfy the equation f^-1(x)=f(x^-1) for f(x)=2x+1:

1. Find f^-1(x) by substituting f^-1(x) into our expression for f, giving us f(f^-1(x))=2f^-1(x)+1.
2. Since f(f^-1(x))=x, this equation is equivalent to x=2f^-1(x)+1, which simplifies to f^-1(x)=(x-1)/2.
3. Assume x solves f^-1(x)=f(x^-1), giving (x-1)/2=2/x+1=(2+x)/x.
4. Cross-multiplying gives x^2-x=4+2x, which simplifies to x^2 - 3x - 4 = 0.
5. Factoring gives (x-4)(x+1)=0, which gives x=4 or x=-1.
6. The sum of the solutions is 4+(-1) = 3, which is found using Vieta's formula that tells us the sum of the roots of a quadratic ax^2+bx+c is -b/a.