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**Topic**

Which of the following statements is correct?

A. The solution set of the equation \(\sqrt {f\left( x \right)} = \sqrt {g\left( x \right)} \) is the solution set of the equation \(f\left( x \) right) = g\left( x \right)\)

B. The solution set of the equation \(\sqrt {f\left( x \right)} = \sqrt {g\left( x \right)} \) is the solution set of the equation \({\left)[ {f\left( x \right)} \right]^2} = {\left[ {g\left( x \right)} \right]^2}\)

C. The solution set of the equation \(f\left( x \right) = g\left( x \right)\) is the solution set of the equation \(\sqrt {f\left( x \right)} = \ sqrt {g\left( x \right)} \)

D. The solution set of the equation \(\sqrt {f\left( x \right)} = \sqrt {g\left( x \right)} \) is the solution set of the equation \(f\left( x \) right) = g\left( x \right)\) satisfying the inequality \(f\left( x \right) \ge 0\) (or \(g\left( x \right) \ge 0\) )

**Solution method – See details**

\(\sqrt {f\left( x \right)} = \sqrt {g\left( x \right)} \Leftrightarrow \left\{ \begin{array}{l}f(x) \ge 0\\ f\left( x \right) = g\left( x \right)\end{array} \right.\)or \(\left\{ \begin{array}{l}g(x) \ge 0\ \f\left( x \right) = g\left( x \right)\end{array} \right.\)

**Detailed explanation**

We have: \(\sqrt {f\left( x \right)} = \sqrt {g\left( x \right)} \Leftrightarrow \left\{ \begin{array}{l}f(x) \ge 0\\f\left( x \right) = g\left( x \right)\end{array} \right.\)

Thus the solution set of the equation \(\sqrt {f\left( x \right)} = \sqrt {g\left( x \right)} \) is the set of solutions of the equation \(f\left( x \) right) = g\left( x \right)\) satisfying the inequality \(f\left( x \right) \ge 0\) (or \(g\left( x \right) \ge 0\) )

Choose D.

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