$\displaystyle f(x)=7(x-4)^{\frac{3}{4}} - 3(x-4)^{\frac{7}{4}}$
Do the y-intercepts exist for this function? According to wolframalpha they do, but I don't see how that's possible.
You are correct in that there are no y-intercepts. Skeeter wants you to write exactly what you entered into Wolfram: somewhere, there must have been a mistake with the way you presented your formula to Wolfram or there was a misinterpretation of the formula on WA's part.
Try writing the exponents in decimal form.