Question

Show f(g(x))=x and g(f(x))=x if f(x)=x^2+1 and g(x)=sqrt x-1

Answer 1

`\bf \begin{cases} f(x) = x^2+1\\ g(x) = √(x-1) \end{cases} \\\\[-0.35em] ~\dotfill\\\\ f(~~g(x)~~)=[g(x)]^2+1\implies f(~~g(x)~~)=[√(x-1)]^2+1 \\\\\\ f(~~g(x)~~)=√((x-1)^2)+1 \\\\\\ f(~~g(x)~~)=x-1+1\implies \blacktriangleright f(~~g(x)~~)=x \blacktriangleleft \\\\[-0.35em] ~\dotfill\\\\ g(~~f(x)~~)=√([f(x)]-1)\implies g(~~f(x)~~)=√([x^2+1]-1) \\\\\\ g(~~f(x)~~)=√(x^2)\implies \blacktriangleright g(~~f(x)~~)=x \blacktriangleleft`