Why is it not the case that f(t, x(t)) = f(t)?

Question

What is the difference between: $f(t)$ and, $f(x(t), t)$

Can I write any function $f(x(t), t)$ as $f(t)$? Why or why not?

Explanations in detail with supporting examples going from simple to complicated would be very helpful.

Answer 1

Writing $f(x(t),t)$ entails $f$ is a function of two variables, while writing $f(t)$ entails it is a function of one variable, so you can never write such an equality. However, it is true that a function of the form $f(x(t),t)$ is of the form $g(t)$, where $g$ is simply defined by $t \mapsto f(x(t), t)$.

Answer 2

Technically, $y=f(t)$ is a function of 1 "free" variable which will "convert" or "map" the given "$t$" to the matching "$y$" , whereas $y=f(x,t)$ is a function of 2 "free" variables.

It may turn out that when we have $y=f(x,t)$ , in some case , $x$ & $t$ may have constraints which make it true that $x=X(t)$ , then we may write $y=f(x,t)=f(X(t),t)$ & see that we have only 1 "free" variable $t$ due to that constraint.

We should now introduce (or make) a new function to write this new condition $y=f(x,t)=f(X(t),t)=F(t)$ where $f$ & $F$ are not same.

It is a (BAD) Practice to sometimes write $f(t)$ (Eg to make it convenient & less verbose on a black board) where rigorous analysis states that we should use $F(t)$ but when there is no ambiguity, we can sometimes get away with this sloppy abuse of notation.

Better avoid this BAD Practice & introduce a new function which is more rigorous & will avoid unnecessary ambiguity & confusion.

OP is asking for Examples, hence consider this :

$z=f(x,y)=|x \cdot y|$ where :
$x,y,z$ are real numbers ,
$x,y$ are Co-Ordinates of Points on a Plane ,
$z$ is the area of the rectangle between the Point $(x,y)$ & the Origin $(0,0)$ with horizontal & vertical sides.

Let us consider only Squares, that is , we want $y=Y(x)=x$
Then we get $z=f(x,Y(x))=|x \cdot Y(x)|=|x \cdot x|=x^2$
We should write it like $z=F(x)=x^2$
We can give a meaning to $F(x)$ that this gives the area of a Square with a side between the Origin & a Point on the x-axis

Here, we see that $f$ was using Points on a Plane while $F$ is using Points on a line.

It is convenient & wrong to write $z=f(x)=x^2$, but it is abuse of notation & will be confusing when overdone. We must be a little verbose to write or introduce the new function correctly.