Question

If \( f\left(1 - \frac{1}{x} \right) = \frac{5x + 1}{x}, \, x \neq 0 \), then \( f(x) = k - x \). What is the value of \( k \)?

• A. 5
• B. 4
• C. 3
• D. 6

Hint:

If a function involves reciprocal transformations, try inverting the expression to find a pattern or direct functional form.

Answer 1

The Correct Option is D

Let \( y = 1 - \frac{1}{x} \Rightarrow x = \frac{1}{1 - y} \). Then, \[ f(y) = f\left(1 - \frac{1}{x} \right) = \frac{5x + 1}{x} = 5 + \frac{1}{x} = 5 + (1 - y) = 6 - y \] Thus, \( f(y) = 6 - y \Rightarrow f(x) = 6 - x \), hence \( k = 6 \). Correct value: 6