is tg^-1 (x) not the same as tg(x)^-1?

Question

Well, for better syntax, my question looks like this:

Isn't $tg^{-1}(x)$ the same as $(tg(x))^{-1}$ ?

I always thought that it was absolutely the same.

But I was trying to solve a task on an edx.org website and I don't understand why their answer is different. Here is the image with the task (and their solution)

In short, it asks what possible values can the following expression have:

$$tg^{-1}(sin(n*\pi)),$$ where n is an integer.

To me it was obvious that $sin(\pi*n)$ is zero, and $tg(0)$ is therefore also zero, and $tg^{-1}(0) \equiv 1/tg(0)$ should be $infinity$.

But in their answer it is said that $tan^{-1}$ function is defined between $-\pi/2$ and $\pi/2$ and "therefore" the answer can only be zero.

Is there something I don't know or am I wrong somewhere?

Answer 1

The notation $\tan^{-1}$ is sometimes used to denote the arctangent function $\arctan$. That is,

$$\tan^{-1}(x) = \arctan(x) = y \Leftrightarrow \tan(y)=x.$$

Answer 2

The notation $$f^{-1}$$ can either mean the inverse of $f$ (when the inverse is defined, sometimes the inverse of a restriction of $f$ to a smaller domain where it is injective), or the point-wise reciprocal of $f$ (when $f$ takes values in a field like $\mathbb{R}$).

The first $f^{-1}$ is with respect to function composition $\circ$, and we have: $$(f^{-1}\circ f) = \mathrm{id} \quad\text{ and } \quad(f\circ f^{-1}) = \mathrm{id}$$ where $\mathrm{id}$ is the identity map. In other words $f^{-1}(f(x))=x$ for all $x$ and $f(f^{-1}(y))=y$ for all $y$.

The second $f^{-1}$ is sometimes written $1/f$, and is with respect to pointwise multiplication, we have for each $x$: $$(f(x))(f^{-1}(x))=1$$ where $1$ is the number one in the codomain of $f$.

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Actually, the same is true for other superscripts (exponents), for example: $$f^3$$ might mean the function $$(f\circ f\circ f)(x) = f(f(f(x)))$$ or the function $$(f\cdot f\cdot f)(x) = (f(x))(f(x))(f(x))$$

For trigonometric functions these confusions are particularly common since many writers use e.g. $\cos^2$ for the point-wise square, and at the same time use $\cos^{-1}$ for the inverse (also known as arcus cosine, arccos).

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See Wikipedia: Inverse trigonometric functions § Notation for a discussion of notations used and their problems.