What is f(x) equal to ?
Consider the following for the next two (02) items that follow : Let 3f(x)+f(\frac{1}{x})=\frac{1}{x}+1
- A. \frac{1}{8x}-\frac{x}{8}+\frac{1}{4}
- B. \frac{3}{8x}-\frac{x}{8}+\frac{3}{4}
- C. \frac{3}{8x}+\frac{x}{8}+\frac{1}{4}
- D. \frac{3}{8x}-\frac{x}{8}+\frac{1}{4} ✓
Correct Answer: D. \frac{3}{8x}-\frac{x}{8}+\frac{1}{4}
Explanation
Given 3f(x) + f(\frac{1}{x}) = \frac{1}{x} + 1. Replace x with \frac{1}{x} to get a second equation: 3f(\frac{1}{x}) + f(x) = x + 1. Multiply the first equation by 3: 9f(x) + 3f(\frac{1}{x}) = \frac{3}{x} + 3. Subtract the second equation from this result: 9f(x) - f(x) = \frac{3}{x} + 3 - (x + 1) \implies 8f(x) = \frac{3}{x} - x + 2. Dividing by 8 gives f(x) = \frac{3}{8x} - \frac{x}{8} + \frac{1}{4}.
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