"you have estimated a linear regression model relating y to x. your professor says, "i think that the relationship between y and x is nonlinear." how would you test the adequacy of your linear regression? "

You can construct a residual plot, plotting the x values against the values of the residuals; if there is a non-linear pattern in the residual plot then it would not be suitable to use linear regression, whilst if there is no noticeable pattern (ie. the points are randomly scattered) then the relationship is linear and a line of best fit could be appropriate.