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Mathematics Question on Strictly increasing or strictly decreasing function

 If f(x)=2(tan⁡−1(ex)−π4), then f(x) is:\text{ If } f(x) = 2\left( \tan^{-1}(e^x) - \frac{\pi}{4} \right), \text{ then } f(x) \text{ is:}

A

even and is strictly increasing in (0,∞)(0, \infty)

B

even and is strictly decreasing in (0,∞)(0, \infty)

C

odd and is strictly increasing in (−∞,∞)(-\infty, \infty)

D

odd and is strictly decreasing in (−∞,∞)(-\infty, \infty)

Answer

odd and is strictly increasing in (−∞,∞)(-\infty, \infty)

Explanation

Solution

Analyze the properties of f(x)=2(tan⁡−1(ex)−π4)f(x) = 2 \left( \tan^{-1}(e^x) - \frac{\pi}{4} \right).

Determine the parity (even or odd) and monotonicity (increasing or decreasing) of f(x)f(x).

Conclude that f(x)f(x) is odd and strictly increasing over (−∞,∞)(-\infty, \infty), verifying option (3) as correct.