Understanding the Definition of arcoshx and Its Role in Inverse Functions

[Gaz031]

This isn't exactly homework but I thought it was too basic to justify putting this post in the general math section.
My question is: why is arcoshx defined as: arcoshx=ln[x+rt(x^2-1)] and not +-ln[x+rt(x^2-1)] ?
Is it simply to keep it as a one to one function? I know that to have an inverse a function should be one to one but why is it not allowed for an inverse function to be 1 to 2 mapping?

 

[Brad Barker]

This isn't exactly homework but I thought it was too basic to justify putting this post in the general math section.
My question is: why is arcoshx defined as: arcoshx=ln[x+rt(x^2-1)] and not +-ln[x+rt(x^2-1)] ?
Is it simply to keep it as a one to one function? I know that to have an inverse a function should be one to one but why is it not allowed for an inverse function to be 1 to 2 mapping?

yeah, it'd be so that the inverse is a function.

in what class/context did you come across this? if you're comfortable with functions of a complex variable, i'd say that it's because you have to specify a particular branch-cut in order for a function to be...nice.

really, with just plain vanilla arccos x, we start off by restricting the domain of cosine, then finding the inverse function of it.

so you can think of the same being done to coshx.

 

[thepatientmental]

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.