Question

The graph of the function \( f(x) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right) \) is symmetric about:

• A. x-axis
• B. y-axis
• C. origin
• D. \( y = x \)

Hint:

To check for symmetry about the origin, verify if \( f(-x) = -f(x) \). If true, the graph is symmetric about the origin.

Answer 1

The Correct Option is C

Step 1: Understand the given function.
The function is \( f(x) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right) \). To analyze the symmetry of the graph, we need to consider how the function behaves under the transformation \( x \to -x \).

Step 2: Check the symmetry about the origin.
For a function to be symmetric about the origin, it must satisfy the property: \[ f(-x) = -f(x). \] Let's evaluate \( f(-x) \): \[ f(-x) = \log_e \left( (-x)^3 + \sqrt{(-x)^6 + 1} \right) = \log_e \left( -x^3 + \sqrt{x^6 + 1} \right). \] Now, compare this with the expression for \( f(x) \): \[ f(x) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right). \] Since \( f(-x) \) and \( f(x) \) are negatives of each other, we can conclude that the graph of \( f(x) \) is symmetric about the origin.

Step 3: Conclusion.
Thus, the graph of the function is symmetric about the origin, and the correct answer is (c).