Question

Which function has an inverse that is also a function?

Answer 1

If an expression is a function, that means all its x-values in the x,y pairs, are never repeated, the y-values can repeat, but not the x-values

now, an inverse expression of a function, in order to be a function, has to also have unique x-values, x-values that never repeat

however, one characteristic of inverse functions is that, their value pairs, are exactly the same as the original function, but flipped-sideways

namely, if a function has a value pair of   3,5  the inverse will have 5,3
if a function has 7, 11  and 13, 27
the inverse has  11, 7  and 27, 13

so... of all those sets above, if any of those is the value pairs of a function, the x-values must be unique, however, if its inverse expression is also a function, then the y-values must also be unique, namely, if you used the flipped-sideways version, x-values must also be unique

so, will be the one whose x-values AND y-values are unique, let's check

$\bf \{(-1, -2), (0, \boxed{4}), (1, 3), (5, 14), (7, \boxed{4})\}\impliedby \textit{has a y-repe}\textit{at, no dice} \\\\\\ \{(-1, \boxed{2}), (0, \boxed{4}), (1, 5), (5, \boxed{4}), (7, \boxed{2})\}\impliedby \textit{has 2 y-repea}\textit{ts, no dice} \\\\\\ \{(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)\} \\\\\\ \{(-1, \boxed{4}), (0, \boxed{4}), (1, 2), (5, 3), (7, 1)\}\impliedby \textit{has a y-repe}\textit{at, no dice}$

Answer 2

Inverse of the function A is not a function because we will have pairs: ( 4, 0 ) and ( 4 , 7 ). Also for the function B : ( 4, 0 ) and ( 4, 5 ).     And for D : ( 4, - 1 ), ( 4, 0 ). We must have 1 value of x for 1 value of y. So an inverse that is also a function is just C.

Answer: C ) { ( - 1 , 3 ) , ( 0 , 4 ), ( 1, 14 ) , ( 5 , 6 ), ( 7 , 2 )}