# Writing correct mathematics -- functions within functions...

1. Jul 22, 2015

### christian0710

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. Jul 22, 2015

### christian0710

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?

3. Jul 22, 2015

### DEvens

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.

4. Jul 23, 2015

### Fredrik

Staff Emeritus
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.

5. Jul 23, 2015

### tommyxu3

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.

6. Jul 24, 2015

### micromass

Staff Emeritus
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.