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

Answer:

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  $H(x)=\frac{x+1}{x-1},\; J(x)=\frac{x+1}{x-1}$. Since H and J are the same function, then

$(H\circ J)(x)=(J\circ H)(x)=J(\frac{x+1}{x-1})=\\=\frac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1}=\frac{\frac{x+1+x-1}{x-1}}{\frac{x+1-x+1}{x-1}}=\frac{2x(x-1)}{2(x-1)}=x$.

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