Discussion
If \(f(x) = x + 1\) , and \(g(x) = x^{3}\) ,then \(f^{-1}(g(x))\) is equal to
Answer: E
\(f^{-1}(g(x))\) is \(x^{3} - 1\).
Since \(f(x) = x + 1\), its inverse undoes the addition: \(f^{-1}(y) = y - 1\). Put \(y = g(x) = x^{3}\).
\[ f^{-1}(g(x)) = x^{3} - 1 \]
\(x^{3} + 2\) applies f a second time instead of its inverse, and \(x - 1\) uses \(f^{-1}(x)\) without composing with g. The other options mix the two functions in other ways. A check at x = 2: \(g(2) = 8\) and \(f^{-1}(8) = 7 = 2^{3} - 1\).
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