[b][color=#bf9000]The relation [math]y=cos\left(x\right)[/math] [/color][/b][b][color=#bf9000]is classified as a function because each and every angle (input) has 1 and only 1 cosine ratio (output.) Consequently, these outputs always range from -1 to 1 (inclusive.) [/color][/b]

No, it is not. In order for a relation to be a function, each input can have only one output. Yet in order for the [color=#cc0000][b]inverse relation[/b][/color] to be a function, [b][color=#1e84cc]each output [/color][/b](of the [color=#bf9000][b]original function[/b][/color]) can only have 1 input that maps to it. Yet this clearly isn't the case for the function [math]y=cos\left(x\right)[/math] because the [color=#1e84cc][b]output[/b][/color] 1/2, for example, has more than one input (angle) (ex's: [math]\frac{\pi}{3}[/math] and [math]\frac{5\pi}{3}[/math] ) that map to it. (In fact, there exist infinitely many angles whose cosine ratio is 1/2.) Therefore, the [color=#cc0000][b]inverse relation[/b][/color] is not a function.

[b]Hint:[/b] Find a piece of the[b][color=#bf9000] original function[/color][/b] over which this function is [b]ALWAYS INCREASING!!! [/b](That way, no [b][color=#1e84cc]output[/color][/b] can have no more than 1 input that maps to it!)