Question

If \( f(x) = \ln \left( \frac{x^2 + e}{x^2 + 1} \right) \), then the range of \( f(x) \) is:

• A. \( (0, 1) \)
• B. \( (0, 1] \)
• C. \( [0, 1] \)
• D. \( \{0, 1\} \)

Hint:

When working with logarithmic functions, consider the behavior of the argument and the natural logarithm's range.

Answer 1

The Correct Option is B

The function is a natural logarithm. While the logarithmic function \( \ln(x) \) has a range of \( (-\infty, \infty) \), the argument of this logarithm is a ratio of two terms, which constrains the function's range based on the values of \( x \). Analysis of the limits and function behavior indicates the range of \( f(x) \) is \( (0, 1] \).