The definition of arcoshx, also known as the inverse hyperbolic cosine function, is closely related to the definition of the hyperbolic cosine function (coshx). Just like how the inverse of a regular cosine function is defined as arccosx, the inverse of coshx is defined as arcoshx. This means that when we input a value into the coshx function, we get an output of that value. Similarly, when we input a value into the arcoshx function, we get an output of that value.
Now, to answer your question about why arcoshx is defined as arcoshx=ln[x+rt(x^2-1)] and not +-ln[x+rt(x^2-1)], we need to understand the concept of inverse functions. Inverse functions are functions that “undo” each other. For example, the inverse of adding 5 to a number is subtracting 5 from that number. They essentially “cancel out” each other’s actions.
In order for a function to have an inverse, it needs to be a one-to-one function, meaning that each input has a unique output. If a function is not one-to-one, then it is not possible to have a unique inverse for each input. This is why the definition of arcoshx only includes the positive value of ln[x+rt(x^2-1)].
But why is it not allowed for an inverse function to be a one-to-two mapping? The main reason is that it would violate the definition of a function. A function must have a unique output for each input, otherwise it would not be considered a function. If a function has two outputs for one input, it is not a function.
In conclusion, the definition of arcoshx is not just to keep it as a one-to-one function, but it is also necessary for it to be a function in the first place. Without this restriction, the inverse of coshx would not be a true inverse and would not have the properties of an inverse function.