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Writing correct mathematics -- functions within functions...

  1. Jul 22, 2015 #1
    Hi I'm a bit confused about some mathematical notation
    If i write

    f(x)=(2x^2 + 10)^4

    And i define
    u= 2x^2 +10
    u^4 = f(x)

    Would it then be correct to write
    f(u)= u^4

    Or would i get
    f(u)= 2(u)^2 +10 = (2(2x^2 +10)+10)^2

    Should i define u^4 = f(x) first? Would it then be correct?

    My book writes that if u is a function of x then then you can put u into x and get the following.
    ∫f(u)du = F(g(x))

    So I want to make sure I'm assuming correctly.
     
  2. jcsd
  3. Jul 22, 2015 #2
    A last question: If you define a function f to have the equation f(x) = 2x is it then true that the variable y=f(x) differs from f, because f is a function (not a number) and y is a number?

    So when we take f(x)= (1+2x)^2 and call u=(1+2x) then it would be wrong to write f=u^2 because f is already defined as a function (not a value?)

    So writing ∫fdx or∫fdf would just be plain wrong?
     
  4. Jul 22, 2015 #3

    DEvens

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    You should keep the form of the function f(x). Like so.
    f(x)=(2x^2 + 10)^4
    f(u)=(2u^2 + 10)^4

    And so on. That is, you should think of the content of the () in f(x) as a placeholder.

    If you want to have u as some function of x then you can define a different function to account for this. Like so.

    u(x) = 2x^2 + 10
    F(u) = u^4
    F(u(x)) = u(x)^4 = (2x^2+10)^4 = f(x)

    Or to use a different example, consider the trig function sin(u). If you plot sin(u) as a function of u for u between 0 and ##2\pi## you get a nice sine wave. If you then think about u being a function of x, say u(x) = x^2, then you have sin(u(x)) = sin(x^2). If you plot this function as a function of x for ##x = 0## to ##\sqrt{ 2\pi}## it will still go through the same values, but the shape will be different. You should try this to see what I mean.
     
  5. Jul 23, 2015 #4

    Fredrik

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    Yes. y is a number because f(x) is a number. However, even though y is a number and not a function, it's still correct to say that y is a function of x, because the value of y is determined by the value of x. You can also say that about f(x). f is a function, but f(x) is a number and a function of x. But don't say that f is a function of x. That wouldn't make sense.

    Yes, this would be wrong. You can write ##f(x)=u^2##, but not ##f=u^2##. However, if you define a function g by g(t)=1+2t for all real numbers t, then you can write ##f=g^2##. This is based on the following definition of the product of two functions: (fg)(x)=f(x)g(x) for all real numbers x. Then we can define the notation ##g^2## by ##g^2=gg##. So ##g^2(x)=(gg)(x)=g(x)g(x)=g(x)^2## for all real numbers x.

    Yes. I think ∫f is an acceptable alternative to the standard ∫f(x)dx, but I still don't recommend using it, because people aren't used to seeing it.
     
  6. Jul 23, 2015 #5
    That would not be wrong but just have different meanings. The point is that you have tell others what you concern, that is, ##dx## and ##df## in this case.
     
  7. Jul 24, 2015 #6

    micromass

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    Yes. The notation ##\int f## is acceptable and ##\int f(x)dx## is acceptable, but not ##\int f dx##. The notation ##\int fdf## is acceptable, but that is the same as ##\int xdx##, which is not what you mean.
     
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