If we write y=f(x)=18/(x-17)+13, then the inverse is expressing x in terms of y.
Multiply through by x-17: y(x-17)=18+13(x-17)=18+13x-221.
xy-17y=13x-203, xy-13x=17y-203, x(y-13)=17y-203, x=(17y-203)/(y-13).
f^-1(x)=(17x-203)/(x-13).
CHECK
f(18)=18+13=31, so f^-1(31) should be 18: (17*31-203)/(31-13)=324/18=18, which is expected.