Definitions

[Dale10101]

Since I am never sure if what I think is true is in fact true without asking I am wondering if I have this right:

A "relation" that passes the "vertical test" is a "function". A function that also passes the "horizontal test" is called a "one-to-one function", aka a "one-to-one correspondence", aka a "bijective function, aka an "unto function", aka an "unto relation".


A function that passes the vertical and horizontal test is "invertible", or more properly perhaps, is an "invertible function" meaning that an "inverse", or more properly, an "inverse function" exists. When a function passes the horizontal test it means that its transposition of variables (dependent to independent) will pass the vertical test and thus be a function, the condition for the original function being invertible.


Although a function may be "invertible", that does not necessarily mean an analytic form of the inverse function exists. For example, as I understand, y = ln(x) + x is an invertible function but no analytic form of the inverse function, x = f(y) can be derived.


Before trying to transpose a function to its inverse function one should graph the function and see if its reflection across the line that bisects the first and third quadrant results in a relation that passes the vertical test, i.e is a function.

Yes/no?

 

[pka]

Since I am never sure if what I think is true is in fact true without asking I am wondering if I have this right: A "relation" that passes the "vertical test" is a "function". A function that also passes the "horizontal test" is called a "one-to-one function", aka a "one-to-one correspondence", aka a "bijective function, aka an "unto function", aka an "unto relation".
A function that passes the vertical and horizontal test is "invertible", or more properly perhaps, is an "invertible function" meaning that an "inverse", or more properly, an "inverse function" exists. When a function passes the horizontal test it means that its transposition of variables (dependent to independent) will pass the vertical test and thus be a function, the condition for the original function being invertible.
Although a function may be "invertible", that does not necessarily mean an analytic form of the inverse function exists. For example, as I understand, y = ln(x) + x is an invertible function but no analytic form of the inverse function, x = f(y) can be derived.
Before trying to transpose a function to its inverse function one should graph the function and see if its reflection across the line that bisects the first and third quadrant results in a relation that passes the vertical test, i.e is a function.

Actually there a great many problems in trying to answer the questions you have posed.

Consider the $\displaystyle \arctan$ function: $\displaystyle \arctan: x\mapsto \arctan (x)$.

Now we know that $\displaystyle \arctan (x)$ maps $\displaystyle \displaystyle\mathbb{R}\to\left(\frac{-\pi}{2},\frac{\pi}{2}\right)$ is a bijection.

However, this is not a bijection $\displaystyle \mathbb{R} \leftrightarrow \mathbb{R}$. But rather is shows that the real numbers are are equally numerous with a bounded open interval.

Thus one must be careful about the claims that one makes about these types of functions.

 

[Dale10101]

OK

Actually there a great many problems in trying to answer the questions you have posed.

Consider the $\displaystyle \arctan$ function: $\displaystyle \arctan: x\mapsto \arctan (x)$.

Now we know that $\displaystyle \arctan (x)$ maps $\displaystyle \displaystyle\mathbb{R}\to\left(\frac{-\pi}{2},\frac{\pi}{2}\right)$ is a bijection.

However, this is not a bijection $\displaystyle \mathbb{R} \leftrightarrow \mathbb{R}$. But rather is shows that the real numbers are are equally numerous with a bounded open interval.

Thus one must be careful about the claims that one makes about these types of functions.

OK, I see what you are saying about a non R -> R correspondence. Also, "But rather is shows that the real numbers are are equally numerous with a bounded open interval." is interesting and gets into that whole question of different types of infinities ... for my purpose I would rather not open that can of worms.

I suppose what I am asking is whether those definitions are what is meant in the context of polynomials, algebraic equations in general or even logarithmic and exponential equations where those words are most likely to be used in this sub forum.

What I am expressing in my question is my understanding of what I am reading in various online tutorials for intermediate students. But maybe my question is too broad or vague and answers will come by presenting the various actual problems as I encounter. Thanks, no worries.


Edit: I am going to simplify and clarify this question and re ask it later, no further response needed for now, thanks.