Question

If \(f(x)=\sqrt{\log_{10}(x^2)}\), the set of all values of \(x\) for which \(f(x)\) is real, is

• A. \([-1,1]\)
• B. \([1,\infty)\)
• C. \((-\infty,-1]\)
• D. \((-\infty,-1]\cup[1,\infty)\)

Hint:

For \(\sqrt{\log(x^2)}\) to be real: \(\log(x^2)\ge 0 \Rightarrow x^2\ge 1\Rightarrow |x|\ge 1\).

Answer 1

The Correct Option is D

Step 1: Condition for square root to be real.
\[ \log_{10}(x^2) \ge 0 \] Step 2: Solve inequality.
\[ \log_{10}(x^2)\ge 0 \Rightarrow x^2 \ge 10^0 = 1 \] Step 3: Solve for \(x\).
\[ x^2 \ge 1 \Rightarrow x \le -1 \ \text{or}\ x \ge 1 \] Final Answer: \[ \boxed{(-\infty,-1]\cup[1,\infty)} \]