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Use the function \(f(x)=x^2+1\) and \(g(x) = \displaystyle\frac{{1}}{{x}}\) to answer the following

  1. Find \(f(g(x))\).
  2. What is the domain of \(f(g(x))\)?
  3. Find \(g(f(x))\).
  4. What is the domain of \(g(f(x))\)?

Solution

  1. \(f(g(x))=\displaystyle\frac{{1}}{x^2}+1\).
  2. Domain of \(f(g(x))\): \((-\infty,0)\cup(0,\infty)\)
  3. \(g(f(x))=\displaystyle\frac{{1}}{x^2+1}\)
  4. Domain of \(g(f(x))\): \((-\infty,\infty)\)

In General: when composing functions, first consider the domain of the inner function, then consider the domain of the result.