Step 1 :Given the functions \(f(x)=\frac{9}{x-4}\) and \(g(x)=\frac{9}{x}+4\)
Step 2 :Substitute \(g(x)\) into \(f(x)\) to find \(f(g(x))\)
Step 3 :\(f(g(x))=f\left(\frac{9}{x}+4\right)=\frac{9}{\left(\frac{9}{x}+4\right)-4}\)
Step 4 :Simplify to get \(f(g(x))=x\)
Step 5 :Substitute \(f(x)\) into \(g(x)\) to find \(g(f(x))\)
Step 6 :\(g(f(x))=g\left(\frac{9}{x-4}\right)=\frac{9}{\left(\frac{9}{x-4}\right)}+4\)
Step 7 :Simplify to get \(g(f(x))=x\)
Step 8 :Since both \(f(g(x))\) and \(g(f(x))\) simplify to \(x\), the functions \(f\) and \(g\) are inverses of each other
Step 9 :\(\boxed{\text{Final Answer:}}\)
Step 10 :a. \(f(g(x))=x\)
Step 11 :b. \(g(f(x))=x\)
Step 12 :c. \(f\) and \(g\) are inverses of each other