Analyzing the Function \( f(x) \) for Odd/Even Properties and Monotonicity

We are given the function \( f(x) = 2 \left( \tan^{-1}(e^x) - \frac{\pi}{4} \right) \) and asked to determine if it is an odd function or an even function, and whether it is strictly increasing or strictly decreasing.

Checking for Odd or Even Function Properties

To check if a function \( f(x) \) is odd or even, we need to evaluate \( f(-x) \) and compare it with \( f(x) \) and \( -f(x) \). A function is odd if \( f(-x) = -f(x) \) for all \( x \) in its domain, and it is even if \( f(-x) = f(x) \) for all \( x \) in its domain.

Let's find \( f(-x) \):

\( f(-x) = 2 \left( \tan^{-1}(e^{-x}) - \frac{\pi}{4} \right) \)

We know that for \( y > 0 \), the identity \( \tan^{-1}(y) + \tan^{-1}\left(\frac{1}{y}\right) = \frac{\pi}{2} \) holds. Since \( e^x > 0 \) for all real \( x \), we have \( \tan^{-1}(e^x) + \tan^{-1}(e^{-x}) = \frac{\pi}{2} \). Thus, \( \tan^{-1}(e^{-x}) = \frac{\pi}{2} - \tan^{-1}(e^x) \).

Substitute this into the expression for \( f(-x) \):

\( f(-x) = 2 \left( \left(\frac{\pi}{2} - \tan^{-1}(e^x)\right) - \frac{\pi}{4} \right) \)

Simplify the expression inside the parenthesis:

\( f(-x) = 2 \left( \frac{\pi}{2} - \frac{\pi}{4} - \tan^{-1}(e^x) \right) \)

\( f(-x) = 2 \left( \frac{2\pi - \pi}{4} - \tan^{-1}(e^x) \right) \)

\( f(-x) = 2 \left( \frac{\pi}{4} - \tan^{-1}(e^x) \right) \)

Factor out -1 from the parenthesis:

\( f(-x) = -2 \left( \tan^{-1}(e^x) - \frac{\pi}{4} \right) \)

We can see that \( -2 \left( \tan^{-1}(e^x) - \frac{\pi}{4} \right) \) is exactly \( -f(x) \).

So, we have found that \( f(-x) = -f(x) \). This confirms that the function \( f(x) \) is an odd function.

Checking for Strict Monotonicity

To determine if a function is strictly increasing or strictly decreasing, we need to find its first derivative, \( f'(x) \), and analyze its sign over the domain. If \( f'(x) > 0 \) for all \( x \) in an interval, the function is strictly increasing in that interval. If \( f'(x) < 0 \) for all \( x \) in an interval, the function is strictly decreasing in that interval.

Let's compute the derivative of \( f(x) \):

\( f(x) = 2 \left( \tan^{-1}(e^x) - \frac{\pi}{4} \right) \)

\( f'(x) = \frac{d}{dx} \left[ 2 \left( \tan^{-1}(e^x) - \frac{\pi}{4} \right) \right] \)

Using the constant multiple rule and the linearity of differentiation:

\( f'(x) = 2 \cdot \frac{d}{dx} \left[ \tan^{-1}(e^x) - \frac{\pi}{4} \right] \)

Using the chain rule for \( \frac{d}{dx}(\tan^{-1}(u)) = \frac{1}{1+u^2} \frac{du}{dx} \), where \( u = e^x \), and the derivative of a constant (\( \frac{\pi}{4} \)):

\( f'(x) = 2 \cdot \left[ \frac{1}{1 + (e^x)^2} \cdot \frac{d}{dx}(e^x) - 0 \right] \)

\( f'(x) = 2 \cdot \left[ \frac{1}{1 + e^{2x}} \cdot e^x \right] \)

\( f'(x) = \frac{2e^x}{1 + e^{2x}} \)

Now, let's analyze the sign of \( f'(x) \). For any real number \( x \), the exponential term \( e^x \) is always positive (\( e^x > 0 \)). The term \( e^{2x} \) is also always positive, so \( 1 + e^{2x} \) is always positive (\( 1 + e^{2x} > 1 \)).

Since the numerator \( 2e^x \) is positive and the denominator \( 1 + e^{2x} \) is positive for all real \( x \), their quotient \( f'(x) = \frac{2e^x}{1 + e^{2x}} \) must be positive for all real \( x \).

\( f'(x) > 0 \) for all \( x \in (-\infty, \infty) \).

Since the first derivative is positive for all \( x \) in its domain \( (-\infty, \infty) \), the function \( f(x) \) is strictly increasing on \( (-\infty, \infty) \).

Conclusion

Based on our analysis, the function \( f(x) = 2 \left( \tan^{-1}(e^x) - \frac{\pi}{4} \right) \) is an odd function and is strictly increasing on the interval \( (-\infty, \infty) \).

Comparing this conclusion with the given options, we find that the function is odd and is strictly increasing in \( (-\infty, \infty) \).

Revision Table: Function Properties and Monotonicity

Additional Information: Inverse Tangent and Exponential Functions

The function \( f(x) \) involves the inverse tangent function, \( \tan^{-1}(y) \), and the exponential function, \( e^x \).

• The domain of \( \tan^{-1}(y) \) is all real numbers \( (-\infty, \infty) \).
• The range of \( \tan^{-1}(y) \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).
• The derivative of \( \tan^{-1}(y) \) with respect to \( y \) is \( \frac{1}{1+y^2} \).
• The function \( e^x \) is defined for all real numbers \( x \in (-\infty, \infty) \).
• The range of \( e^x \) is \( (0, \infty) \).
• The derivative of \( e^x \) with respect to \( x \) is \( e^x \).
• The combination \( \tan^{-1}(e^x) \) is defined for all real \( x \) because the range of \( e^x \) (\( (0, \infty) \)) is within the domain of \( \tan^{-1}(y) \) (\( (-\infty, \infty) \)).
• The range of \( e^x \) is strictly positive, which is why we could use the identity \( \tan^{-1}(y) + \tan^{-1}(1/y) = \pi/2 \) for \( y=e^x > 0 \).