The Practical Usages of Finding the Inverse of a Function

[christian0710]

I see how i caused confusion for my self and others now previously.
Thank's to your patience and effort and my effort, I actually think that it's all clear now.
This way of describing it is also more logic for myself.
This will be my last post, thank you again. I really appreciate it :)

Here is the mathematical correct arguments:_
1. I define the functions and the variables/domain of function: "let f and g be 2 functions defined by the equations"

f(x)=2x+2
g(x)=(x-2)/2

"and let x be defined as all real numbers"So If i EVER again in a forum, write 2 equations
like f(x)=y --> (implies) g(y)=x

Then i should ALWAYS have started out defining 3 things first.
1) the function f and g that the 2 equations are defined by and the domain "for all real numbers x" (also the range?
2) the equations f(x)=2x+2 g(x)=(x-2)/2
3) y belongs to all real numbers.Now I can conclude that

f(x)=y --> this implies
g(y)=x if f and g are inverse, or this implies that f and g are inverse.

f(x)=f(g(y)= because g(y)=x
g(y)=g(f(x)) because f(x)=y

And

f(g(x)) = x
g(f(x)) = x

when i plot the graph of the 2 equations

g(y)=x
f(x)=y

The 2 equations plotted in the same xy-plane
will give the same graph.

if x=1 then

f(1)=3 so the point (x,y)=(1,3)
g(3)=1 so the point (y,x)=(3,1)
The order of the points might be switched, but y=y and x=x
so we get the same points if choose the same xy cordinate
system for both equations.2. If i want to talk about inverse FUNCTIONS (not equation), then I always

need to define the functions (which i already did above, just doing it again for the exercise) and not just the equations.
1) The function f, is defined by the equation f(x)=2x+2,
x belongs to All real numbers.
2) The function g, is defined by the euqation
g(x)=(x-2)/2, x belongs to all real numbers.

Now i know that whatever variable i plug into g or f,

x,b,a,c, apples, bananas they all belong to "all real

numbers", because i just defined that the domain of the

function is all real numbers.

However when i define 2 equations like

g(y)=x
f(x)=y

The equation f(x)=y is still the equation of the function f

(or maybe there is a better way of saying it?) , because if

f(x)=y then y=2x+2, and that's how we defined f for x

belonging to all real numbers.

BUT the function g, is not defined as g(y)=x
because (and i hope this is the right argument - a bit unsure)

1) g is defined by g(y)=(y-2)/2 or it could be called g(x)=(x-2)/2 as long as the variable belongs to all real numbers.

2) we have not defined what x is? Or this might be the right argument sinse x is defined as all real numbers.

Now back to the stuff I'm sure about:

I could use any dummy variable and say g(b)=(b-2)/2 and

this equation would be defined as the function of g, for b defined as all real numbers.

Conclusion
So the whole point is of this topic is , 2 functions f and

g are inverse functions if the 2 equations f(x)=y and g(y)

=x have the same graph, and this is not always the case.

if f(x)=x^2+ 2
if g(x)=x+3

f(x)= x^2+2
g(f(x))=(x^2+2)^3 +3 so they are not equivalent and so the

functions f and g are not inverse.

And yes purplemath.com was correct because g(x)=(x-2)/2 is the same as g(y)=(y-2)/2 (for the domain defined as all real numbers)
But if we now define y=f(x) then the meaning of g(y) changes into g(f(x)) and this is not = (y-2)/2 , but = x YESS :D

Thank you so much guys!

 

[christian0710]

My advice would be to stop thinking like this! You're taking relatively simple aspects of maths and twisting and turning them round until you've lost the meaning in all the convolutions. It seems to me that there is analysis that leads to clarity and simplification and analysis that leads to confusion and complexity, and you are indulging in the latter.

If you over-analyse anything for long enough, you lose the meaning. Check out "semantic satiation":

https://en.wikipedia.org/wiki/Semantic_satiation

Yes and no, because my analysis led me to understand things i couldn't understand before, and it led me to be more aware of how carefully i need to think about how to define and describe a problem :)

 

[Mark44]

But if you consider a function to be defined as a collection of pairs, then we get $f$={$(x,y):y=f(x)$}, and $f^{-1}$={$(y,x): y=f(x)$} This shows that when plotted, both on the xy-axis, they will have the same graph, but, if you accept this definition of function, they are not the same function.

Of course they're not the same function (with only the identity function excepted). No one is claiming that a function and its inverse are the same function.

 

[Helolo]

Hi, I know that the inverse function of

y= f(x) =2x+1

is

y-1=2x
x=(y-1)/2

and then we just replace x with f^-1(y) and then when we plug in any value of y it gives us a corresponding x value.

Now my question is this: If we want to find a line or function that is perpendicular to another line or function, then we do the same steps to go from y=2x+1 to x=(y-1)/2 and then we switch x and y to get
y=(x-1)/2

Why is it practical to find an inverse of a function if they both have the same graph? If you ploty y=2x+1and x=(y-1)/2 you get the same graph, so what are the practical usages for finding the inverse of a function?
Is the function that is perpendicular to another function, also a kind of an inverse even thought we switch x and y?

Wrong, if $y = f(x) = 2x + 1$ then the inverse function should be $f^{-1}(2x+1) = x$. $y = f(x) \Longleftrightarrow x = f^{-1}(y)$.