I know that they exist for exponential functions (we currently have them in class), but to me it doesn't seem "reasonable" when I look at the definition of what an inverse function is. The inverse is defined as a function where you can swap $x$ and $y$, then solve for $y$ and the notation being $\operatorname{f^{-1}}(x)$. Since functions are a 1 to 1 mapping this can only be true for some functions. In the textbook we use we have following definition for the domain of functions/inverse functions:
$$\mathbb{D}_{f} = \mathbb{W}_{f^{-1}} \rightleftharpoons \mathbb{W}_{f} = \mathbb{D}_{f^{-1}}$$
I also get that some functions don't have inverses or where they only exist for a restricted domain (like $x^2$ where you have to restrict the domain, or some functions where you can't solve for $x$).
The thing about as example $2^x$ that puts me off is that the input domain $\mathbb{D}$ consists out of all real numbers, whereas the output is made out of positive real numbers only. How can there be a 1 to 1 mapping if the output consists only out of positive real numbers, aren't there less positive real numbers than real numbers? With as example $x^3$ you use up all $x$ and $y$ values, so it having a valid inverse makes intuitive sense to me. We get taught about how important the uniqueness of the mapping between $x$ and $y$ is, but it just feels wrong for exponential functions.
Can anyone provide me a pointer to as where I start thinking about this wrongly? I have solved all the problems in our book and on the additional sheet the teacher gave us and have only had a few mistakes (which probably came from lack of sleep). Understanding the composition of functions was pretty easy for me as well, thanks to knowing higher order functions. I'm really sure I'm misunderstanding something elementary the wrong way.
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$\begingroup$To make sense of the situation, we have to rethink what it means for two sets of numbers to have "the same amount of elements."
The function $f(x)=x^3$, as you mention, associates each real number $x$ with exactly one other real number, $y=x^3$. In this case, $f:\mathbb{R}\rightarrow\mathbb{R}$ gives a correspondence between the real numbers and itself.
What was important was that we had a one-to-one correspondence through a function. But the example of the exponential shows that we can find a correspondence between the real numbers and a different set, the positive numbers $(0,\infty)$. For each real number $x$, we associate it to the positive number $2^x$. The reverse correspondence, coming from the inverse function of the function $f(x)=2^x$, is that we associate each positive number $y$ with the real number $\log_2(y)=x$. Since $f$ is invertible, each real number goes to one, unique positive number under $f$, and each positive number goes to one, unique real number under $f^{-1}$.
This process of finding an invertible function between two sets of numbers in a one-to-one fashion is a way to make sense of "having the same amount of elements" for two sets. This particular type of association is given the name cardinality.
A different way of answering the question might be to say that any set with infinitely many elements should have "the same amount of elements." However, this definition does not play nicely with the context of invertible functions. For example, there is no invertible function from the natural numbers $\mathbb{N}$ to the real numbers $\mathbb{R}$ that associates each real number with a unique natural number, even though both are infinite. (We can easily send a natural number $n$ to that same number as an element of $\mathbb{R}$, but there is no way to go in the reverse direction from each real number to one, unique natural number. See Cantor's diagonal argument.)
The upshot is that invertible functions give one way of identifying two sets, the domain and range of an invertible function (where here by range I mean the image of $f$ and not its codomain, as the function must be what we call surjective.) They are not the same set, but they do happen to have a form of correspondence between them through the exponential/logarithm. Two sets have the same cardinality when there is at least one function providing such a correspondence.
You can now compare the example of natural numbers and even numbers. The even numbers are a subset of natural numbers, but you can associate each even number with the unique natural number which is half its value, and each natural number $n$ with the even number $2n$. Here, the function is $f(n)=2n$ with inverse $f^{-1}(m)=\frac{m}{2}$. One set lies within the other, but there happens to exists a correspondence through the function $f$ between the sets.
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