Question:
If \( f(x) = \ln \left( \frac{x^2 + e}{x^2 + 1} \right) \), then the range of \( f(x) \) is:
If \( f(x) = \ln \left( \frac{x^2 + e}{x^2 + 1} \right) \), then the range of \( f(x) \) is:
Show Hint
When working with logarithmic functions, consider the behavior of the argument and the natural logarithm's range.
Updated On: Feb 15, 2025
- \( (0, 1) \)
- \( (0, 1] \)
- \( [0, 1] \)
- \( \{0, 1\} \)
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
The function is in the form of a natural logarithm. We know that the logarithmic function \( \ln(x) \) has a range of \( (-\infty, \infty) \), but since the argument of the logarithm is a ratio between two terms, the range of the function is constrained by the values of \( x \). After evaluating the limits and checking the behavior of the function, we determine that the range of \( f(x) \) is \( (0, 1] \).
Was this answer helpful?
0
0
Top Questions on Limit and Continuity
- If \( f(x) = \left(\frac{1+x}{1-x}\right)^{\frac{1}{x}} \) is continuous at \( x = 0 \), then \( f(0) \) is:
- AP EAMCET - 2024
- Mathematics
- Limit and Continuity
- Let \( f(x) \) be defined as: \[ f(x) = \begin{cases} 0, & x = 0 \\ 2 - x, & 0 < x < 1 \\ 2, & x = 1 \\ 1 - x, & 1 < x < 2 \\ -\frac{3}{2}, & x \geq 2 \end{cases} \] Then which of the following is true?
- AP EAMCET - 2024
- Mathematics
- Limit and Continuity
- Evaluate: \[ \lim_{x \to 0} \left[ \frac{1}{x} - \frac{1}{e^x - 1} \right] \]
- AP EAMCET - 2024
- Mathematics
- Limit and Continuity
The limit: \[ \lim_{x \to 0} \frac{\sin \left( \pi \sin^2 x \right)}{x^2} \] is equal to:
- KEAM - 2024
- Mathematics
- Limit and Continuity
- Evaluate the limit: \[ \lim_{x \to 4} \left( \frac{1}{x - 4} - \frac{5}{x^2 - 3x - 4} \right) \] is equal to
- KEAM - 2024
- Mathematics
- Limit and Continuity
View More Questions
Questions Asked in VITEEE exam
- At what temperature should a gold ring of diameter 6.230 cm be heated so that it can be fitted on a wooden bangle of diameter 6.241 cm? Both the diameters have been measured at room temperature (27 °C).
Given: Coefficient of linear thermal expansion of gold \( \alpha = 1.4 \times 10^{-5} \, \text{K}^{-1} \). - A cylinder of height 20 m is completely filled with water. The velocity of efflux of water (in ms\(^{-1}\)) through a small hole on the side wall of the cylinder near its bottom is:
- The resistivity (\( \rho \)) of a semiconductor varies with temperature. Which of the following curves represents the correct behavior?
- JEE Main - 2023
- VITEEE - 2024
- Electrostatics
- The refractive index \( \mu \) of the material of a prism is given by: % Formula for Refractive Index \[ \mu = \frac{\sin \left( \frac{A + \delta_m}{2} \right)}{\sin \left( \frac{A}{2} \right)} \] where \( A \) is the apex angle of the prism and \( \delta_m \) is the angle of minimum deviation.
- The output \( Y \) of the following logic circuit for given inputs is:
- VITEEE - 2024
- Thermodynamics
View More Questions