Question

Compute the compositions H ∘ J and J ∘ H to determine whether or not J and H are inverses. (Simplify your answers completely.) H and J are both defined from ℝ − {1} to ℝ − {1} by the formula H(x) = x + 1 x − 1 , J(x) = x + 1 x − 1 for each x ∈ ℝ − {1}

Answer 1

Remember, if H(x) and J(x) are functions, then
`(H\circ J)(x)=H(J(x))`. And G(x) is the inverse function of H(x) if
`(G\circ H)(x)=(H\circ G)(x)=x`

With
`H(x)=(x+1)/(x-1),\; J(x)=(x+1)/(x-1)`. Since H and J are the same function, then

`(H\circ J)(x)=(J\circ H)(x)=J((x+1)/(x-1))=\\=((x+1)/(x-1)+1)/((x+1)/(x-1)-1)=((x+1+x-1)/(x-1))/((x+1-x+1)/(x-1))=(2x(x-1))/(2(x-1))=x`.

Since
`(J\circ H)(x)=(H\circ J)(x)=x`, then H and J are inverses.