How to Prove an Inverse Function Using Equating Square Roots?

[#neutrino]

if

then to prove an inverse of this exists the following has been done to show that it is one to one

what is the basis of equating the 2 square roots ?

 

[Simon Bridge]

Can you think of another way to show it is 1-1?

 

[Mark44]

if
View attachment 94299 then to prove an inverse of this exists the following has been done to show that it is one to one

View attachment 94298

what is the basis of equating the 2 square roots ?

What is the definition of a function being one-to-one?

 

[#neutrino]

What is the definition of a function being one-to-one?

an output can have only one input ,but what i don't understand is the basis of the expression

 

[#neutrino]

Can you think of another way to show it is 1-1?

yes i can reach the conclusion if we draw a graph but I'm confused about how to arrive at the conclusion (one to one ) using this data

 

[jbriggs444]

yes i can reach the conclusion if we draw a graph but I'm confused about how to arrive at the conclusion (one to one ) using this data

If you graph it, can you think of a specific feature of the graph that you could phrase mathematically?

 

[HallsofIvy]

an output can have only one input ,but what i don't understand is the basis of the expressionView attachment 94307

That is exactly what you said above: "an output can have only one input". If x and y are the "inputs" for the "outputs", f(x) and f(y), and they are the same, f(x)= f(y), so they are the same output, the inputs must be the same: x= y. Showing that "if f(x)= f(y) then x= y" is exactly the same as showing "an output can have only one input".

 

[FactChecker]

an output can have only one input ,but what i don't understand is the basis of the expressionView attachment 94307

It is the "If" part of an "if-then" statement. It is not proven, it is assumed. So there is no need for a "basis" for the statement.
If (x/(x+1))^0.5 = (y/(y+1))^0.5
then x=y.

This is all that you need to prove to show that the function has an inverse.

 

[#neutrino]

That is exactly what you said above: "an output can have only one input". If x and y are the "inputs" for the "outputs", f(x) and f(y), and they are the same, f(x)= f(y), so they are the same output, the inputs must be the same: x= y. Showing that "if f(x)= f(y) then x= y" is exactly the same as showing "an output can have only one input".

It is the "If" part of an "if-then" statement. It is not proven, it is assumed. So there is no need for a "basis" for the statement.
If (x/(x+1))^0.5 = (y/(y+1))^0.5
then x=y.

This is all that you need to prove to show that the function has an inverse.

so why have you proved an input equals an input ? when what we should prove is that for two particular inputs the OUTPUT will be the same only if those two inputs are equal.

 

[FactChecker]

That is how you prove it. Prove that if the result of the function is the same, then the inputs were the same. That what "or two particular inputs the OUTPUT will be the same only if those two inputs are equal." means. So your question is more about how to phrase the logic rather than about function inverses. This might be a good, simple example of using truth tables to see that the logic is correct.