Text Given f(x)=1x−1f(x)=\dfrac{1}{x-1}f(x)=x−11, find the simplified (f∘f)(x)(f\circ f)(x)(f∘f)(x), its domain and compute (f∘f)(3)(f\circ f)(3)(f∘f)(3). Solution Substitute f(x)f(x)f(x) for xxx: (f∘f)(x)=1f(x)−1=11x−1−1=11−(x−1)x−1=x−12−x.(f\circ f)(x)=\frac{1}{f(x)-1}=\frac{1}{\dfrac{1}{x-1}-1}=\frac{1}{\dfrac{1-(x-1)}{x-1}}=\frac{x-1}{2-x}.(f∘f)(x)=f(x)−11=x−11−11=x−11−(x−1)1=2−xx−1. We need x≠1x\ne 1x=1 (for fff) and x≠2x\ne 2x=2 (for the second application): D=R∖{1,2}D=\mathbb{R}\setminus\{1,2\}D=R∖{1,2}. (f∘f)(3)=3−12−3=2−1=−2.(f\circ f)(3)=\frac{3-1}{2-3}=\frac{2}{-1}=-2.(f∘f)(3)=2−33−1=−12=−2. (f∘f)(x)=x−12−x, D=R∖{1,2},(f∘f)(3)=−2 \boxed{\,(f\circ f)(x)=\dfrac{x-1}{2-x},\ D=\mathbb{R}\setminus\{1,2\},\quad (f\circ f)(3)=-2\,}(f∘f)(x)=2−xx−1, D=R∖{1,2},(f∘f)(3)=−2