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dyn
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Hi.
If I have a function f(x) = √(x+1) and I define u=x+1 is it correct to state f(u) = √u ?
If I have a function f(x) = √(x+1) and I define u=x+1 is it correct to state f(u) = √u ?
No. If ##u=x+1## then ##x=u-1## and you get ##f(x)=f(u-1)=\sqrt{u}=\sqrt{x+1}\,.##dyn said:Hi.
If I have a function f(x) = √(x+1) and I define u=x+1 is it correct to state f(u) = √u ?
##f(u)=\sqrt{u+1}##. The name of the variable doesn't matter. You could as well write ##f(tree)=\sqrt{tree+1}##, but this would be a bit confusing and too long to use.dyn said:Thanks , what would f(u) be then ?
Again, no. ##f(u) = \sqrt{u + 1}##, as already explained by @fresh_42.dyn said:Consider f(z) = √ (z+1) , now define u = z+1 then f(u) = √u ,
This part isn't relevant to what you asked about.dyn said:this is then used to show that f(u) is multi-valued around u=0 and so f(z) is multi-valued around z= -1
dyn said:Hi.
If I have a function f(x) = √(x+1) and I define u=x+1 is it correct to state f(u) = √u ?
dyn said:No , there was no integral. Another similar example is differentiation using the chain rule , for example if
##y(x) = (x+1)^2## then using u=x+1 , ##y(u) = u^2## and then this is differentiated using the chain rule
I do not think so. As you see in the picture, there are two different curves. Now if you only consider their slopes, then they behave the same (at different points). If you integrate them, then the results will be the same (within different integration limits). In neither case it is a matter of notation. They were and will be two different functions. This is especially important for physicists, as the frames are important here! A mathematician could say "I don't care, I'm only interested in the geometric object", but a physicist cannot. Change of coordinates is completely uninteresting for mathematicians, physicists do nothing else! So it is not an abuse of notation, it is a lack of description from your part. Your initial question is a strict NO, and then you came up with the chain rule, which didn't make any sense without further context. ##f(u)=\sqrt{u+1}## if ##u=x## and ##f(u-1)=\sqrt{u}## if ##u=x+1##. There is literally nothing which can be abused! ##g(u)=\sqrt{u}## is a different function, in physics as in mathematics. Fullstop.dyn said:Thanks for all your replies. I think the problem is the difference between the way physicists and mathematicians are strict/not strict about notation
I was quite happy with the answers before this post but the above post does not make any sensefresh_42 said:I do not think so. As you see in the picture, there are two different curves. Now if you only consider their slopes, then they behave the same (at different points). If you integrate them, then the results will be the same (within different integration limits). In neither case it is a matter of notation. They were and will be two different functions. This is especially important for physicists, as the frames are important here! A mathematician could say "I don't care, I'm only interested in the geometric object", but a physicist cannot. Change of coordinates is completely uninteresting for mathematicians, physicists do nothing else! So it is not an abuse of notation, it is a lack of description from your part. Your initial question is a strict NO, and then you came up with the chain rule, which didn't make any sense without further context. ##f(u)=\sqrt{u+1}## if ##u=x## and ##f(u-1)=\sqrt{u}## if ##u=x+1##. There is literally nothing which can be abused! ##g(u)=\sqrt{u}## is a different function, in physics as in mathematics. Fullstop.
All what came after post #2 (or #4) is pure guesswork (including mine, with the exception of this one) based on lacking context, information and clarity.
Changing the argument of a function refers to modifying the input value that is passed into the function. This can affect the output or result of the function, making it a useful tool for manipulating data or performing different calculations.
To change the argument of a function, you can simply modify the value that is passed into the function when it is called. This can be done by assigning a new value to the variable or parameter that represents the argument, or by using a different value altogether.
Yes, changing the argument of a function can definitely affect its return value. Since the argument is used as the input for the function, modifying it can change the way the function processes the data and ultimately result in a different return value.
There are many reasons why one might want to change the argument of a function. Some common reasons include performing different calculations or operations on the data, filtering or manipulating the data in a specific way, or customizing the output of the function based on different inputs.
Yes, there can be limitations to changing the argument of a function. Depending on the function's design, there may be specific data types or formats that are required for the argument, or certain constraints on the range of values that can be used. Additionally, changing the argument may not always result in a meaningful or useful output, so it is important to understand the function's purpose and how it processes data before making any modifications.