Question

Find the range of \( f(x) = \log_e(4x^2 - 4x + 1) \)

• A. \( (-\infty, 0] \)
• B. \( (0, \infty) \)
• C. \( (-\infty, \infty) \)
• D. \( [0, \infty) \)

Hint:

The range of a logarithmic function is determined by the values of the argument. Since logarithms are defined for positive arguments, the range will depend on the behavior of the function inside the logarithm.

Answer 1

The Correct Option is A

We need to determine when the argument of the logarithm is positive. For the function \( f(x) = \log_e(4x^2 - 4x + 1) \), the quadratic expression inside the logarithm is always positive, as it is a perfect square: \[ 4x^2 - 4x + 1 = (2x - 1)^2 \] Since \( (2x - 1)^2 \geq 0 \), the argument of the logarithm is always positive. The range of \( f(x) \) is determined by the behavior of the logarithmic function. Since the argument can take any positive value, the logarithm can take any value from \( -\infty \) to \( 0 \), inclusive. Therefore, the range of \( f(x) \) is: \[ (-\infty, 0] \]