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The arctangent function, often written as \( \arctan x \) or \( \tan^{-1} x \), is one of the inverse trigonometric functions. Its principal value lies in the range \( -\frac{\pi}{2} < y < \frac{\pi}{2} \) which makes it unique compared to other trigonometric functions. Understanding these properties is crucial when dealing with calculations and transformations involving \( \arctan \).

When you input any real number into \( \arctan \), it provides the angle (in radians) whose tangent is that number. It's worth noting that because \( \tan \) is not defined for \( \pm \frac{\pi}{2} \) (or \( 90 \) degrees), the range of \( \arctan \) does not include these endpoint values. Furthermore, \( \arctan \) function has an odd symmetry, meaning \( \arctan(-x) = -\arctan(x) \). This property helps in understanding how the function behaves under transformations and when analyzing its graph.

The range of an inverse trigonometric function is crucial for understanding how it maps real numbers to angles. Inverse trigonometric functions, including \( \arctan \), take a real number and return an angle, serving as the inverse operations to their respective trigonometric functions.

For instance, as previously mentioned, the range of \( \arctan \) is \( -\frac{\pi}{2} < y < \frac{\pi}{2} \), strictly excluding the endpoints. This is different from \( \arcsin \) which has a range of \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \) and \( \arccos \) with a range of \( 0 \leq y \leq \pi \). These ranges ensure that each real number corresponds to a unique angle, preserving the function鈥檚 one-to-one nature.

Understanding this concept is key when determining the range of composite functions involving inverse trigonometric operations, such as the function given in the exercise \( f(x)=2 \tan^{-1}(1-x^2)+\frac{\pi}{6} \).

Function transformations involve shifting, stretching, compressing, or reflecting a function's graph. There are a few basic types of transformations:

• Vertical shifts move the graph up or down.
• Horizontal shifts move the graph left or right.
• Vertical stretches and compressions multiply the function鈥檚 output by a factor.
• Horizontal stretches and compressions divide the input by a factor before applying the function.
• Reflections flip the graph over a specified axis.

In the context of the given exercise, we see multiple transformations applied to the \( \arctan \) function.

Initially, the function \( \tan^{-1}(1-x^2) \) is manipulated by stretching it vertically by a factor of 2, altering its range. Subsequently, a vertical shift is applied by adding \( \frac{\pi}{6} \), which adds a constant value to all the y-values of the function.

Such transformations are vital in altering the range of functions, as demonstrated in the solution. Each step in the transformation process alters the range of the function, which ultimately determines \( f(x) \)鈥檚 range to be \( -\frac{5\pi}{6} \leq f(x) \leq \frac{7\pi}{6} \).