Define H: R - {1} -> R - {1} and J: R - {1} -> R - {1} as follows:
H(x) = J(x) = (x + 1) / (x - 1) for each x in R - {1}
Compute the compositions H β J and J β H to determine if J and H are inverses for each other. (Simplify your answers completely.)
For every x in R - {1},
(H β J)(x) = H(J(x)) = H((x + 1) / (x - 1))
= (((x + 1) / (x - 1)) + 1) / (((x + 1) / (x - 1)) - 1)
= 2x / 2
= x
Now use the definition to find J β H.
For every x in R - {1}, (J β H)(x) = J(H(x)) = x
Thus, both compositions equal the identity function on R - {1}, and so H and J are inverses for each other.