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Inverse trigonometric functions, such as the inverse tangent function (\( \tan^{-1} x \)), play a crucial role in understanding function ranges. The inverse tangent function maps any real number to a range between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), which means that no matter the input, the output will always be a value within this interval.
This range becomes significant when transformations, such as squaring the inverse tangent, occur. By squaring, we shift from potential negative and positive outputs to strictly non-negative outputs. Hence, if \((\tan^{-1}x)^2\), the range becomes \(0 \leq y_1 < \frac{\pi^2}{4}\).
This concept ensures that even if the input x changes, the squared output only fluctuates within this interval. Thus, inverse trigonometric transformations expand our understanding of how inputs can be compressed or rearranged into specific output intervals.

Algebraic functions, such as \( \frac{2}{\sqrt{1+x^2}} \), are formed using simple algebraic operations involving powers, roots, or simple arithmetic operations. In this example, the division and root operations create a function that is always positive when x is a real number.
The function \( \frac{2}{\sqrt{1+x^2}} \) achieves various values based on x. As |x| increases towards infinity, the denominator increases, so the function's value drops, approaching zero. At x = 0, the function achieves its maximum, which is 2.
Thus, the range is expressed as \(0 < y_2 \leq 2\). This property is vital because it informs us how algebraic functions can invariably maintain positive ranges despite varying inputs.

A combination of functions involves merging two or more functions into one, such as \(f(x)=(\tan^{-1}x)^2 + \frac{2}{\sqrt{1+x^2}}\). When studying combinations, it’s essential to find the individual range of each function first, and then consider how they interact.
In this function, each component contributes different aspects to the final range. \((\tan^{-1}x)^2\) contributes values ranging from 0 to \(\frac{\pi^2}{4}\), while \(\frac{2}{\sqrt{1+x^2}}\), fluctuates between 0 (as |x| approaches infinity) and 2.
The range of the total function \(f(x)\) can be determined by adding the smallest and the largest values produced by each part: 0 (from both going to infinity) to \(\frac{\pi^2}{4} + 2\). This demonstrates how combining functions can cover a unique range that is a result of their independent contributions and mutual interactions.