Thread: Quick question about inverse functions.

Under what conditions is the inverse of a function a function?

Can/does this have something to do with the input and outputs of a function or is it as simple whether or not it passes the vertical line test?

Thanks!


-Tom

Hello, Tom!

Under what conditions is the inverse of a function a function?

Can/does this have something to do with the input and outputs of a function
or is it as simple whether or not it passes the vertical line test?

It is as simple as the vertical line test.


There's an "eyeball" approach, too.

Look at the graph of the function f(x).
If it passes the "horizontal line test", its inverse is a function.
(That is, if all horizontal lines intersect the graph in at most one point,
. . then the inverse is also a function.)

Thank you!

Originally Posted by Soroban

Hello, Tom!


It is as simple as the vertical line test.


There's an "eyeball" approach, too.

Look at the graph of the function f(x).
If it passes the "horizontal line test", its inverse is a function.
(That is, if all horizontal lines intersect the graph in at most one point,
. . then the inverse is also a function.)

Here's another quick question... So an example of that would be the functions (1,2) , (3,4). That's a function. The inverse is (2,1) , (4,3) and that's a function too. Am I getting this correctly?

Thanks,


-Tom

Originally Posted by TomCat

Here's another quick question... So an example of that would be the functions (1,2) , (3,4). That's a function. The inverse is (2,1) , (4,3) and that's a function too. Am I getting this correctly?

Thanks,


-Tom

those are coordinates.

a function is an equation, such as "f(x)=(1/2)x+5"

Originally Posted by TomCat

Here's another quick question... So an example of that would be the functions (1,2) , (3,4). That's a function. The inverse is (2,1) , (4,3) and that's a function too. Am I getting this correctly?

Thanks,


-Tom

Yes. A function (more formally but not completely) is a set of ordered pairs. Sometimes you can infinite many like (x,x^2) or sometime finite like here. Thus, yes it is a function and that is its inverse.

Originally Posted by ThePerfectHacker

Yes. A function (more formally but not completely) is a set of ordered pairs. Sometimes you can infinite many like (x,x^2) or sometime finite like here. Thus, yes it is a function and that is its inverse.

really? Do you mean that any two points can have a function, or that any two points are a function.

Originally Posted by Quick

really? Do you mean that any two points can have a function, or that any two points are a function.

A single point can define a function.
For example, {(1,1)}

There is a formal definition of a function that appeared in the 19th Century due to one of my favorites, Dirichlet. Ever since then people started using his definition. A single point fits this definition.

The first time the word "function" was used by the German polymath/mathematician Gottfiend von Leibniz (he was as much as a nemesis to Newton as I am to CaptainBlank). He definied informally as a relationship between two things. For example, the size of a tree as a function of time.

When LaTeX is back online I can give you a formal treatement of what a function is in a mathematical sense. (But be warned this is going to be your first tour of pure mathematics. It might be dangerous).

Originally Posted by ThePerfectHacker

When LaTeX is back online I can give you a formal treatement of what a function is in a mathematical sense.

See you in two weeks