Function Misconceptions


\(f\) and \(g\) are two functions such that \((f \circ g)(3)=5\)

What is the value of \((g \circ f)(3)\) ?

If \(f\) is a function with domain \(\mathbb{R}\) and range \(\mathbb{R}\), does there exist a function \(g\) such that \(g(x) = f^{-1}(x)\) ?

In the above image, the red graph is a graph of \(y=f(x)\), for some function \(f\).

What transformation of \(f\) will produce the blue graph in the above graph?

\(f(x)\) is a function with domain \(\mathbb{R}.\)

\(g(x)\) is a horizontal transformation of \(f(x)\) such that \(g(x) = f(x+k).\)

\(h(x)\) is a vertical transformation of \(f(x)\) such that \(h(x) = f(x)+k.\)

If you choose \(f(x)\) carefully, is it possible that \(g(x)\) might equal \(h(x)\) for all \(x\)?

\(f\) is a function whose domain is \(\mathbb{R}\).

\(g\) is a function such that \(g(x)=f^{-1}(x)\)

True or false: The domain of \(g\) must also be \(\mathbb{R}\).


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