Discussion
Let \(f(x) = x^{2} + 2x + 3, x \leq -1\) . Then the domain of the inverse of \(f(x)\) is
Answer: E
The domain of the inverse is \([2, \infty)\).
The domain of \(f^{-1}\) is the range of \(f\). Complete the square:
\[ f(x) = x^{2} + 2x + 3 = (x + 1)^{2} + 2 \]
For \(x \leq -1\), \((x + 1)^{2}\) takes every value from 0 (at \(x = -1\)) upward, so \(f(x)\) takes every value from 2 upward. On this domain \(f\) is one-one (decreasing), so the inverse exists, with domain \([2, \infty)\). The other intervals do not start at the minimum value 2.
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