Why in some textbooks, the range of $\text{ArcCosh}(x)$ is given $[0, \infty)$?

Question

I have a simple question. Why in some textbooks, the range of $\text{ArcCosh}(x)$ is reported as $[0, \infty)$? Why do they ignore the negative part?

Answer 1

Because of two things:

One:

$\mathrm{arccosh}$ is defined as the inverse of the bijective function

$$f: [0,\infty)\to\mathbb [1, \infty)\\ f: x\mapsto \cosh(x)$$

Two:

If $g$ is the inverse of a bijective function $f:A\to B$, then the range of $g$ is always $A$.

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Therefore: In order for $\mathrm{arccosh}$ to have a range of $\mathbb R$, the it would have to be an inverse of a function from $\mathbb R$. But $\cosh$ is not bijective on $\mathbb R$, so the range of $\mathrm{arccosh}$ cannot be $\mathbb R$.

Answer 2

$\cosh$ is a bijection from $\mathbb{R}_+$ to $[1,+\infty)$. Its reciprocal $\arccos$ is therefore defined on $[1,+\infty)$ and has range $\mathbb{R}_+$.