christian0710 said:
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.
christian0710 said:
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".
christian0710 said:
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}##.
christian0710 said:
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##.
christian0710 said:
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".
christian0710 said:
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.
christian0710 said:
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.
christian0710 said:
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.
christian0710 said:
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.