Problem 4
A point in the plane with a Cartesian coordinate system is called a mixed point if one coordinate is rational and the other is irrational. Find all polynomials with real coefficients whose graphs contain no mixed point.
Step 7 of 7: Verify and conclude
Detailed analysis
A nonconstant linear polynomial with rational coefficients maps rational inputs to rational outputs. If its output were rational at an irrational input, solving x=(f(x)-b)/a would make the input rational, impossible. Constants fail because an irrational input then gives a mixed point. Thus these and only these polynomials work.