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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}

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Answer:

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

With  [tex]H(x)=\frac{x+1}{x-1},\; J(x)=\frac{x+1}{x-1}[/tex]. Since H and J are the same function, then

[tex](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[/tex].

Since [tex](J\circ H)(x)=(H\circ J)(x)=x[/tex], then H and J are inverses.

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