Question

Consider the real-valued function \(f(x) = \log \left( \frac{3x - 7}{\sqrt{2x \times 2x}} \right) - 7x + f\). Find the domain of f(x).

• A. (-∞, 7/3)
• B. R - (3/2, 2)
• C. (7/3, ∞)
• D. R - {3/2, 2, 7/3}
• E. R - {7/3}

Answer 1

The Correct Option is C

To determine the domain of the function \( f(x) = \log \left( \frac{3x - 7}{\sqrt{2x \times 2x}} \right) - 7x + f \), let's first understand the constraints imposed by the logarithmic function and other components.

1. Logarithmic Condition: The expression inside the logarithm must be positive, i.e., \( \frac{3x - 7}{\sqrt{2x \times 2x}} > 0 \).
2. Simplification: The expression becomes: \(\sqrt{2x \times 2x} = 2x\), so the inequality is: \(\frac{3x - 7}{2x} > 0\)
3. Sign Change Analysis:
   • For the fraction to be positive, both numerator and denominator should have the same sign.
   • Case 1: \(3x - 7 > 0\) and \(2x > 0\). This implies \(x > \frac{7}{3}\) and \(x > 0\), which leads to \(x > \frac{7}{3}\).
   • Case 2: \(3x - 7 < 0\) and \(2x < 0\). This implies \(x < \frac{7}{3}\) and \(x < 0\), which cannot occur simultaneously since \(\frac{7}{3}\) and \(x < 0\) cannot satisfy both conditions.
4. Domain: The domain is the set of all \(x\) that satisfy the condition \(x > \frac{7}{3}\).
5. Conclusion: Therefore, the domain of \(f(x)\) is \((7/3, \infty)\), confirming that the correct option is (7/3, ∞)

Answer 2

To determine the domain of the function \(f(x) = \log \left( \frac{3x - 7}{\sqrt{2x \times 2x}} \right) - 7x + f\), we need to consider the points where the expression inside the logarithm is positive:

\(1. \; 3x - 7 > 0 \Rightarrow x > \frac{7}{3}\) 

\(2. \; \sqrt{2x \times 2x} \neq 0 \Rightarrow x \neq 0\)

The square root term \(\sqrt{2x \times 2x} = 2|x|\). Since \(x\) is positive for \(x > \frac{7}{3}\), this becomes simply \(2x\), and we need the numerator to be greater than zero while the denominator remains non-zero:

\(\frac{3x - 7}{2x} > 0\)

Simplifying gives:

\(3x - 7 > 0 \Rightarrow x > \frac{7}{3}\)

Both conditions imply \(x > \frac{7}{3}\).

Thus, the domain of \(f(x)\) is the set of all real numbers \(x\) such that \(x > \frac{7}{3}\).

Therefore, the domain is \((\frac{7}{3}, \infty)\).