The Practical Usages of Finding the Inverse of a Function

[christian0710]

Okay let's try again

if f is a function with the equation defined for all x,
f(x)=y= 2x+1

Then

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

is an equivalent equation defined for all y: so x=g(y) and y=f(x) have the same graph.

This is not the same equation or graph as
g(x)=(x-1)/2 defined for all x,
Because f(g(x)) = 2(x-1)/2+1 =x we can say that g and f are inverse,If i plot f(x)=2x+1 and g(x)= (x-1)/2 then they are mirror images of the line y=x.

 

[WWGD]

Okay let's try again

if f is a function with the equation defined for all x,
f(x)=y= 2x+1

Then

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

is an equivalent equation defined for all y: so x=g(y) and y=f(x) have the same graph.

This is not the same equation or graph as
g(x)=(x-1)/2 defined for all x,
Because f(g(x)) = 2(x-1)/2+1 =x we can say that g and f are inverse,If i plot f(x)=2x+1 and g(x)= (x-1)/2 then they are mirror images of the line y=x.

Strictly speaking, this is incorrect: f(x)=2x+1 describes a line of slope 2 in the xy-plane. g(y)=x=(y-1)/2 is a line of slope 1/2 in the yx-plane.

So you need to define things more clearly, when you say they are the same graph. I think you may be looking for functions f with f(f(x))=x, of which f(x)=x is an example.

 

[christian0710]

Strictly speaking, this is incorrect: f(x)=2x+1 describes a line of slope 2 in the xy-plane. g(y)=x=(y-1)/2 is a line of slope 1/2 in the yx-plane.

So you need to define things more clearly, when you say they are the same graph. I think you may be looking for functions f with f(f(x))=x, of which f(x)=x is an example.

Okay so the image i had in mind was that i would plot both functions in the xy plane, and that that g(y) takes in the outputs from f(x), and gives back the inputs that f(x) takes.. How would i discribe this more correctly then?

 

[christian0710]

However, when defining the function f and its inverse, you can do so with any dummy variable, as long as the domain and range are either known or implied.
Example, if you have a function defined by the relation f(v)=2vf(v) = 2v and its inverse g(a)=a/2 g(a) = a/2 then defining y = f(x) implies that y = 2x.
You might quickly be able to see that if y=f(x), then g(y) = g(f(x))= (2x)/2= x.
Notice that if you were to plot these functions, (x,f(x) ) would generate points like (0,0), (1,2), (2,4)... and (g(x),x) would generate points like (0,0), (1/2,1), (1,2) ...

Thank you, this helped me a a lot :)

So I'm going to define that v=f(x) so now i can say that f(-1)(v)= x and because i defined v=f(x) f^(-1)(f(x))=x

Then the set of points for f will be (x,y)=(x,f(x))=f(-1)(v),v) Okay so if fx

f(m)=22m

then it's inverse is

s(b)=b/22 Could i also write?

s(b)=f(m)/22 - do people often define y=f(x) to avoid having to write f(x) because it's easier to just write "y"? if

s(b)=b/22

and

f(m)=22m

and i define y=s(t) so not instead of b, just change the dummy variable to t. Smart :)
So if i define y=f(x) for all x, and f(x)=2x+1
and i definy x=g(y) for all y, and g(y) =(y-2)/1
Then because i defined that g only takes inputs of y, which are the outputs of y=f(x), is it then correctly asumed that g(y)=(y-2)/1 is not the same as
g(x) = (x-2)/1 if x is defined for all x, or what exactly is the correct argument that makes g(y) and g(x) different.

 

[Fredrik]

Okay let's try again

if f is a function with the equation defined for all x,
f(x)=y= 2x+1

Then

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

is an equivalent equation defined for all y: so x=g(y) and y=f(x) have the same graph.

Your statements are still very hard to decode. Does the "for all x" refer to what you wrote before those words, or what you wrote after? I mean, are you saying that f is a function "with the equation defined for all x", or are you saying that for all x, we have f(x)=y=2x+1? If it's the former, I don't know what that means. My guess would be that you're saying that the domain of f is $\mathbb R$. If it's the latter, then what is the symbol y doing in a "for all x" statement?

Here's how I would say what I think you're trying to say:

Let f be the function defined by f(t)=2t+1 for all real numbers t. Let g be the function defined by g(t)=(t-1)/2 for all real numbers t. For all real numbers x and y, if y=f(x) then x=(y-1)/2=g(y).

This is not the same equation or graph as
g(x)=(x-1)/2 defined for all x,

The statement is strangely worded, but if you're trying to say that f and g don't have the same graph, you are correct.

Because f(g(x)) = 2(x-1)/2+1 =x we can say that g and f are inverse,

f(g(x))=x for all x in the domain of g may not be enough. In general, you need f(g(x))=x for all x in the domain of g, and g(f(x))=x for all x in the domain of f.

Example: Define g by $g(t)=\sqrt{t}$ for all positive real numbers t. Define f by $f(t)=t^2$ for all real numbers t. For all x in the domain of g, we have $f(g(x))=(\sqrt{x})^2 =x$, but g can't be the inverse of f since
$$g(f(-1))=\sqrt{(-1)^2}=\sqrt 1=1\neq -1.$$

If i plot f(x)=2x+1 and g(x)= (x-1)/2 then they are mirror images of the line y=x.

If f is such that f(x)=2x+1 for all real numbers x, and g is such that g(x)=(x-1)/2 for all real numbers x, then yes, the graphs of f and g are each other's reflections in the line y=x.

Strictly speaking, this is incorrect: f(x)=2x+1 describes a line of slope 2 in the xy-plane. g(y)=x=(y-1)/2 is a line of slope 1/2 in the yx-plane.

But he didn't write g(y)=x=(y-1)/2, which would suggest that the values of x and y are still constrained by the equation y=f(x). He wrote g(x)=(x-1)/2, which suggests that it's a "for all x" statement.

 

[Mark44]

Strictly speaking, this is incorrect: f(x)=2x+1 describes a line of slope 2 in the xy-plane. g(y)=x=(y-1)/2 is a line of slope 1/2 in the yx-plane.

We normally speak of the x-y plane, in that order, not the y-x plane. The equations y = 2x + 1 = f(x) and x = (y - 1)/2 = g(y) both have exactly the same graph in the x-y plane. Both graphs represent exactly the same line.

So you need to define things more clearly, when you say they are the same graph.

I disagree. IMO, the OP was pretty clear.

I think you may be looking for functions f with f(f(x))=x, of which f(x)=x is an example.

 

[WWGD]

We normally speak of the x-y plane, in that order, not the y-x plane. The equations y = 2x + 1 = f(x) and x = (y - 1)/2 = g(y) both have exactly the same graph in the x-y plane. Both graphs represent exactly the same line.
I disagree. IMO, the OP was pretty clear.

But when y is the independent variable and x depends on y, it seems to make more sense to talk about the y-x plane. In this respect the lines are different, in that , e.g., they have different slopes. EDIT, of course, if you have y=f(x), which has a global inerse x=f(y) , then the two are exactly the same function, "from different perspectives". Maybe this is a reasonable definition.

 

[christian0710]

Your statements are still very hard to decode. Does the "for all x" refer to what you wrote before those words, or what you wrote after? I mean, are you saying that f is a function "with the equation defined for all x", or are you saying that for all x, we have f(x)=y=2x+1? If it's the former, I don't know what that means. My guess would be that you're saying that the domain of f is R\mathbb R. If it's the latter, then what is the symbol y doing in a "for all x" statement?

Here's how I would say what I think you're trying to say:

Let f be the function defined by f(t)=2t+1 for all real numbers t. Let g be the function defined by g(t)=(t-1)/2 for all real numbers t. For all real numbers x and y, if y=f(x) then x=(y-1)/2=g(y).

Okay, I this part I was trying to write exactly what you wrote in your first post "The graph of the function f defined by f(t)=2t+1 for all t, "

So I'm guessing its mathematically incorrect to say
1. An equation is defined for all t? I guess i need to define that t∈R
2. A function is defined for all t?

What I'I'm trying to say is that the domain of the function f with the euqation f(x)= 2x+2 is x ∈ R. Then I want theese 2 equations to be different like you did in your first post: I want this g(y)=(2-y)/2 I to be different from this g(x)=(2-x)/2

I feel discouraged by my lack of mathematical jargon, so I think I'll go back to reading the book and solving problems, I think It's better i pick up a book on mathematical language some time to save time and further misunderstanding.

 

[christian0710]

Let f be the function defined by f(t)=2t+1 for all real numbers t. Let g be the function defined by g(t)=(t-1)/2 for all real numbers t. For all real numbers x and y, if y=f(x) then x=(y-1)/2=g(y).

Yes This makes sense,

So my analisis of what you did
1. You define a function f by the equation f(t))2t+1 by all real numbers t, then you define a second function g by the equation g/(t)=(t-1)/2 for all real numbers t.

2. Now you say y=f(x) so you set a variable y equal to f(x) that depends on x. .
3. Why don't you have to define what x is? Like x ∈R
4 Now could also call it f(u) it does not matter right`?, if you call it f(u) then y=f(u)=2u+2. so if this is true that y=2u+1 then u=g(y) so g(y)=(y-1)/2 =u

So now we have y=f(u) and u=g(y) so now in the xy plane theese 2 graphs are the same, because y=f(u) is the output value of the function f, and this output value is the input value of the function g.

So what argument would you use to turrn g with the equation g(y)=u into the inverse function of f that is reflected as a mirror in the line y=x in the yx plane?

My confusion is this: If f is defined by g(t)=2t+1 for all real numbers of t and the function g is defined by g(t)=(t-1)/2 for all real numbers of t, then g and f are inverses (I agree) but if y=f(u) and u=g(y), then f and g are still inverse, but the graphs are equivalent.

 

[Fredrik]

So I'm guessing its mathematically incorrect to say
1. An equation is defined for all t? I guess i need to define that t∈R

I wouldn't say that an equation like y=2x+1 is "defined for all x". I might say that the variables x and y represent real numbers, and that the values of these variables are constrained by the equation y=2x+1. But I'm more likely to say that x and y are real numbers such that y=2x+1.

However, that's not what I was concerned about. When I said that "if it's the former, I don't know what that means" I was referring to the part of the sentence that said "f is a function with the equation defined for all x". This doesn't make sense since you hadn't yet specified the equation. When I said that "if it's the latter, then what is the symbol y doing in a 'for all x' statement", the concern was that "for all x, y=2x+1" is a false statement regardless of the value of y. For example, it's not the case that all real numbers x are such that 2x+1=3.

I think it would have helped if you had just split the sentence in two.

2. A function is defined for all t?

That's OK, if the scope of the "for all" is clear from the context (in this thread, it's clear that the scope is $\mathbb R$ unless we say otherwise). However, I think the following options are better.

1. The domain of f is $\mathbb R$.
2. f is defined for all real numbers.
3. f(t) is defined for all real numbers t.

I feel discouraged by my lack of mathematical jargon, so I think I'll go back to reading the book and solving problems, I think It's better i pick up a book on mathematical language some time to save time and further misunderstanding.

I realize that this is frustrating, but one good way to learn this is to just try to express yourself as clearly as possible, and allow other people to point out the things that aren't clear. Another thing you can do is to study proofs. When you've studied a proof in a book, try to prove the theorem for yourself without looking at the book. Then go through your proof several times to see if there's room for improvement. (You'll be surprised to see how many improvements there are to make). Put it aside for at least half an hour, and then see if you still think that what you've written is easy to follow.

I don't know if there's a book that explains how to write like a mathematician. But there are some books like "Book of proof" by Richard Hammack, that are meant to teach students how to prove things. Such a book could be useful.

 

[christian0710]

Sory I just realized i was editing this post as you were online. Here is my analysis of what you wrote. I think this is as clear as i can get :)

So my analisis of what you did
1. You define a function f by the equation f(t))2t+1 by all real numbers t, then you define a second function g by the equation g/(t)=(t-1)/2 for all real numbers t.

2. Now you say y=f(x) so you set a variable y equal to f(x) that depends on x. .
3. Why don't you have to define what x is? Like x ∈R
4 Now could also call it f(u) it does not matter right`?, if you call it f(u) then y=f(u)=2u+2. so if this is true that y=2u+1 then u=g(y) so g(y)=(y-1)/2 =u

So now we have y=f(u) and u=g(y) so now in the xy plane theese 2 graphs are the same, because y=f(u) is the output value of the function f, and this output value is the input value of the function g.

So what argument would you use to turrn g with the equation g(y)=u into the inverse function of f that is reflected as a mirror in the line y=x in the yx plane?

My confusion is this: If f is defined by g(t)=2t+1 for all real numbers of t and the function g is defined by g(t)=(t-1)/2 for all real numbers of t, then g and f are inverses (I agree) but if y=f(u) and u=g(y), then f and g are still inverse, but the graphs are equivalent.

 

[christian0710]

That's OK, if the scope of the "for all" is clear from the context (in this thread, it's clear that the scope is R\mathbb R unless we say otherwise). However, I think the following options are better.

1. The domain of f is R\mathbb R.
2. f is defined for all real numbers.
3. f(t) is defined for all real numbers t.

Okay, let's see if i somewhat got this right:
First 3 correct ways to define a function which all state the same.

1. The domain of f is R
2. f is defined for all real numbers
3. f(t) is defined for all real numbers t

But wit a second!
If i can say f(t) is defined for all real numbers t, then what does this mean? I'm clearly not saying that the function f is defined for all real numbers? 2."for all x, y=2x+1" makes no sense, becuase if y=2 then x is not all real numbers, then x must be 1/2. So instead i should say "for all x and all corresponding values of y beloning to R "?
Or maybe just say "the function f defined by f(x)=2x+1 for all real numbers x?

3. There is a difference between writing f(x)=2x and y=2x if you have not defined y=f(x).

4. If i define "f is a function with the equation f(x)=2x for all real numbers of x" is this correct? If not, then what is corrct?

5. If f is a function defined by f(x)=2x for all real numbers x and g is a function defined by g(x)=2x for all real numbers x, then the 2 functions are inverse. BUT if i define u=f(t) then g(u)=t Right? And then the euqations are equivalent, because g(u)=t, takes the output of f and spits out the input of f, right?

6. is the function of g defined by the equation g(x)=(x-1)/2 for all real numbers x belonging to the doman R, the same as function as the function for the the equation u=g(y) in the above sample where y=f(u) ?

. But there are some books like "Book of proof" by Richard Hammack

Thank you, and thak you for your patience, I'll get that book, I think I will need it if i have to pose further questions in this forum to save myself and other people time.

 

[Fredrik]

So my analisis of what you did
1. You define a function f by the equation f(t))2t+1 by all real numbers t, then you define a second function g by the equation g/(t)=(t-1)/2 for all real numbers t.

2. Now you say y=f(x) so you set a variable y equal to f(x) that depends on x. .
3. Why don't you have to define what x is? Like x ∈R

For a sentence that contains a variable to be a statement, i.e. something that's either true or false, one of the following must be true:

1. The variable has previously been assigned a value. (In other words, we have specified what the symbol represents).
2. The variable is the target of a "for all".
3. The variable is the target of a "there exists".

I used option 2. The statement was "for all real numbers x and y, if y=f(x) then g(y)=x". So I was saying that the implication
$$y=f(x)\ \Rightarrow\ g(y)=x$$ holds regardless of what real numbers the symbols x and y represent at the moment.

4 Now could also call it f(u) it does not matter right`?, if you call it f(u) then y=f(u)=2u+2. so if this is true that y=2u+1 then u=g(y) so g(y)=(y-1)/2 =u

Right. A variable that's the target of a "for all" can be replaced with any other variable; this doesn't change the meaning of the statement. So I could have said e.g. "for all real numbers u and x, if y=f(u) then g(y)=u". I could also have used the symbol u in the definition of f, like this: "Let f be the function such that f(u)=2u+1 for all real numbers u".

So now we have y=f(u) and u=g(y) so now in the xy plane theese 2 graphs are the same, because y=f(u) is the output value of the function f, and this output value is the input value of the function g.

So what argument would you use to turrn g with the equation g(y)=u into the inverse function of f that is reflected as a mirror in the line y=x in the yx plane?

My confusion is this: If f is defined by g(t)=2t+1 for all real numbers of t and the function g is defined by g(t)=(t-1)/2 for all real numbers of t, then g and f are inverses (I agree) but if y=f(u) and u=g(y), then f and g are still inverse, but the graphs are equivalent.

What argument would I use to turn g into the inverse function of f? It doesn't need to be turned into the inverse, because the definition ensures that it is the inverse. To prove that, it's sufficient to prove that f(g(t))=t for all t in the domain of g, and g(f(t))=t for all t in the domain of f.

If we just replaced the symbol x with u in some "for all x" statements, then the meaning of the statements are still the same. So I'm not sure I understand what it is about the use of u instead of x that confuses you. I think a lot of what you've been confused about in this thread comes from the following: There's no ambiguity about what's meant by the graph of a function, but there is some ambiguity about what's meant by the graph of an equation. For example, you can associate at least two different sets with the equation y=2x+1: $\{(x,y)\in\mathbb R^2|y=2x+1\}$ and $\{(y,x)\in\mathbb R^2|y=2x+1\}$. When someone speaks of "the" graph of the equation y=2x+1, they always mean the former, because it's conventional to let x be the first component of the pair.

I guess your book would call $\{(x,y)\in\mathbb R^2|y=2x+1\}$ the graph of the equation in the xy-plane, and $\{(y,x)\in\mathbb R^2|y=2x+1\}$ the graph of the equation in the yx-plane. I wouldn't use this terminology, because it suggests that the xy-plane and the yx-plane are different sets, when they're in fact both the set $\mathbb R^2$.

If you want to avoid the ambiguity, you can simply stop talking about graphs of equations. You can still say things like this: The set $\{(x,y)\in\mathbb R^2|y=2x+1\}$ is the graph of a function, and we denote this function by f. The set $\{(y,x)\in\mathbb R^2|y=2x+1\}$ is the graph of $f^{-1}$.

Okay, let's see if i somewhat got this right:
First 3 correct ways to define a function which all state the same.

1. The domain of f is R
2. f is defined for all real numbers
3. f(t) is defined for all real numbers t

But wit a second!
If i can say f(t) is defined for all real numbers t, then what does this mean? I'm clearly not saying that the function f is defined for all real numbers?

Statement 3 means that the string of text "f(t)" has been assigned a value (i.e. we have specified what number it represents) for each real number t. To define a function f is to specify the domain of f, and to specify what f(t) is for all t in the domain. That means that to say that f(t) is defined for all t in $\mathbb R$, is to say f has been defined and that is domain is a set that has $\mathbb R$ as a subset.

So 3 and 1 aren't saying exactly the same thing (oops, that was unintentional). 3 is saying that the domain of f is a set that contains $\mathbb R$.

2."for all x, y=2x+1" makes no sense, becuase if y=2 then x is not all real numbers, then x must be 1/2. So instead i should say "for all x and all corresponding values of y beloning to R "?

You want the "for all" part of the statement to specify exactly what values of x and y are allowed in the statement that comes next, but you're letting the statement that comes next specify what the allowed values of y are. The statement I suggested doesn't have that problem: "for all real numbers x and y, if y=f(x), then g(y)=x".

3. There is a difference between writing f(x)=2x and y=2x if you have not defined y=f(x).

Yes. It's kind of hard to explain what that difference is. I'd say that to interpret f(x)=2x as the definition of a function, you only have to think of an appropriate "for all" statement (for each real number x, we define f(x)=2x). But to interpret y=2x as the definition of a function, you have to think of it as a statement about the variables themselves, rather than as a statement about real numbers. The string of text "y=2x" specifies which assignments of values to x and y are allowed. It does so in a way that ensures that the value of y can be computed from the value of x, and that the value of x can be computed from the value of y. That means that it indirectly defines two functions, one of which is the inverse of the other.

4. If i define "f is a function with the equation f(x)=2x for all real numbers of x" is this correct? If not, then what is corrct?

It's OK. The "of" shouldn't be there, but that's probably just something you missed in editing. I would however prefer to say that "f is a function such that f(x)=2x for all real numbers x", or "f is the function defined by f(x)=2x for all real numbers x". (Note that these statements don't say exactly the same thing. The former allows the domain of f to be a larger set that has $\mathbb R$ as a subset).

You could also say that "f is the function with domain $\mathbb R$ defined by the equation y=2x", but this is a little bit ambiguous, as discussed above.

5. If f is a function defined by f(x)=2x for all real numbers x and g is a function defined by g(x)=2x for all real numbers x, then the 2 functions are inverse. BUT if i define u=f(t) then g(u)=t Right? And then the euqations are equivalent, because g(u)=t, takes the output of f and spits out the input of f, right?

Yes.

6. is the function of g defined by the equation g(x)=(x-1)/2 for all real numbers x belonging to the doman R, the same as function as the function for the the equation u=g(y) in the above sample where y=f(u) ?

This is a good example of how a seemingly insignificant detail can completely change the meaning of the statement. Presumably, you meant "for all $x\in\mathbb R$", but the "belonging to" changes the sentence so that it says that the function g is a real number.

You seem to be asking if the function g defined by g(x)=(x-1)/2 for all real numbers x, is equal to the function that's indirectly defined by the equation u=g(y), where y=f(u). I don't see how the "where" fits into this, because if we use y=f(u), the equation that's supposed to define a function becomes u=g(f(u)), which doesn't define a function in any obvious way. I think you probably meant to ask about the function indirectly defined by the equation u=(y-1)/2. As I said earlier in this post, an equation that has a unique solution defines two functions. In this case, one of them is g. The other is f.

Since you wrote u=(y-1)/2 instead of the equivalent y=2u+1, I would guess that the function you have in mind is the one whose graph is $\{(y,u)|y\in\mathbb R, ~u=(y-1)/2\}$. That function is equal to g.

 

[PeroK]

Okay, let's see if i somewhat got this right:
First 3 correct ways to define a function which all state the same.

1. The domain of f is R
2. f is defined for all real numbers
3. f(t) is defined for all real numbers t

But wit a second!
If i can say f(t) is defined for all real numbers t, then what does this mean? I'm clearly not saying that the function f is defined for all real numbers? 2."for all x, y=2x+1" makes no sense, becuase if y=2 then x is not all real numbers, then x must be 1/2. So instead i should say "for all x and all corresponding values of y beloning to R "?
Or maybe just say "the function f defined by f(x)=2x+1 for all real numbers x?

3. There is a difference between writing f(x)=2x and y=2x if you have not defined y=f(x).

4. If i define "f is a function with the equation f(x)=2x for all real numbers of x" is this correct? If not, then what is corrct?

5. If f is a function defined by f(x)=2x for all real numbers x and g is a function defined by g(x)=2x for all real numbers x, then the 2 functions are inverse. BUT if i define u=f(t) then g(u)=t Right? And then the euqations are equivalent, because g(u)=t, takes the output of f and spits out the input of f, right?

6. is the function of g defined by the equation g(x)=(x-1)/2 for all real numbers x belonging to the doman R, the same as function as the function for the the equation u=g(y) in the above sample where y=f(u) ?
Thank you, and thak you for your patience, I'll get that book, I think I will need it if i have to pose further questions in this forum to save myself and other people time.

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

 

[WWGD]

We normally speak of the x-y plane, in that order, not the y-x plane. The equations y = 2x + 1 = f(x) and x = (y - 1)/2 = g(y) both have exactly the same graph in the x-y plane. Both graphs represent exactly the same line.
I disagree. IMO, the OP was pretty clear.

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.

 

[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 $f^{-1}(2x+1) = x$. $y = f(x) \Longleftrightarrow x = f^{-1}(y)$.