Combinatorics › Sets and Venn diagrams › Counting relations and functions

Value missing from range of a Möbius function

High school · ●●●○○ 50 pts

The function ff is defined by f(x)=ax+bcx+df(x)= \frac{ax+b}{cx+d}, where aa, bb, cc and dd are nonzero real numbers. It has the properties f(19)=19f(19)=19, f(97)=97f(97)=97 and f(f(x))=xf(f(x))=x for all xx except −dc-\frac{d}{c}. Find the unique number that is not in the range of ff.

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functional equationsinverse functionsrange of a functiondifference of squares