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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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    $\begingroup$ “Aren’t there less positive real numbers than real numbers?” Well, no. At least, not if by “how many” you mean “can be put into one-to-one correspondence.” Just like there are just as many positive integers as there are even positive integers (consider the map sending $n$ to $2n$), in the same manner we can put the positive real numbers into a one-to-one correspondence with all real numbers. $\endgroup$ – Arturo Magidin May 15 '20 at 20:24
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    $\begingroup$ What a really insightful question! "Aren't there less positive real numbers than real numbers?" No! $\endgroup$ – Miguel May 15 '20 at 20:24
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    $\begingroup$ As the other commenters have mentioned already, the issue is with your idea that "aren't there less positive real numbers than real numbers?" No. Infact, you can put the real numbers $\Bbb{R}$ in one-to-one correspondence with the interval $(-\pi/2, \pi/2)$ (or if you want, the interval $(-1,1)$). For example, by considering the $\arctan$ function from $\Bbb{R} \to (-\pi/2, \pi/2)$. $\endgroup$ – peek-a-boo May 15 '20 at 20:28
  • $\begingroup$ Thank you @ArturoMagidin, this makes more sense. A friend of mine has told me about the map from $n$ to $2n$ but it wasn't as clear to me initially (your answer is more concise, but it does make my brain hurt.. a little). $\endgroup$ – spmlzz May 15 '20 at 20:34
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    $\begingroup$ Once it stops hurting, look up “cardinality”. It’s a mind-bending concept, but interesting. A good place to start is Episode 7 of the podcast “A Brief History of Mathematics” with Marcus du Satoy. You can find it here $\endgroup$ – Arturo Magidin May 15 '20 at 20:38
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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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  • $\begingroup$ Would a proof that no two different $x$ values map to the same $y$ be like the following? $$2^{x_1} = 2^{x_2} \\ \log_{2} 2^{x_1} = \log_{2} 2^{x_2} \\ x_1 \cdot \log_{2} 2 = x_2 \cdot \log_{2} 2$$ because $\log_{2} 2$ is one $x_1 = x_2$? Thanks for the very elaborate answer, I've started looking up some stuff and it makes a lot more sense. $\endgroup$ – spmlzz May 15 '20 at 23:33
  • $\begingroup$ @spmlzz: That argument has the potential of being circular, because the existence of the logarithmic function may depend on a proof that the exponential function is one-to-one in the first place! In fact, one has a choice. You can either define the exponential function, then prove it is one-to-one (on the basis of the definition), and then use that fact to define logarithms; or you can define the logarithmic function, then prove it is one-to-one (on the basis of the definition), and then use that fact to define the exponential function. $\endgroup$ – Arturo Magidin May 16 '20 at 4:14

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