1. Understanding y=y(x)

Hello,
i am having trouble understanding the notation y=y(x). I understand the functional notation y=f(x).

I have been trying to study implicit functions and this notation is used quite regularly. What would be the best reference? Could someone please provide some insight?

Thank you

2. Re: Understanding y=y(x)

You probably understand this just fine, it's just shorthand.

If y=f(x) then y(x)=f(x). But I might use y later like a variable as in

$y^2 + \sin y -\sqrt{y}=10$

the above line just means

$(f(x))^2 + \sin (f(x)) - \sqrt{f(x)} = 10$

and probably more importantly

$y'(x) = \frac{dy}{dx}=\frac{df}{dx}=f'(x)$

Is it any clearer now?

3. Re: Understanding y=y(x)

A function f is a rule which assigns a value (dependent) to one or more values (independent).
Ex f= sin, cos, square, sqrt, etc.

y=f(x) says y is the value f assigns to x.

y alone can stand for either a dependent or independent variable.
y=y(x) says y depends on x and is a shortcut which says y is the function and the value of the function.

Sometimes shortcut can lead to confusion:
y=f(x), x=g(z), y=f(g(z)) = F(z)
y=y(x), x=x(z), y=y(x(z)) =? y(z). y in y(z) is not the same function as y in y(x). You have to keep track.

Implicit functions:

x+2y=3 defines y as a fuction of x or x as a function of y.
x=3-2y, dx/dy=-2
y=3/2-x/2, dy/dx=-1/2

or, think of x and y as functions of t:
dx/dt+2dy/dt=0 and multiply through by dt:
dx+2dy=0 and solve for dx/dy or dy/dx.

y=x2
dy=2xdx
dy/dx=2x
dx/dy=1/2x=1/2y1/2

or, f(x,y)=0 -> df=0 and solve for dx/dy or dy/dx

4. Re: Understanding y=y(x)

Normally, this convention comes up in the study of differential equations: that is, equations involving functions and their derivatives.

Normally, we have a function, f, defined on a domain X, and which takes values in Y, as in f: X-->Y.

Often, both X and Y are the "same kind of thing", as is the case when both equal the set of real numbers. It is then common to represent f as a subset of the plane (a GRAPH) consisting of those points in the plane:

(x,y) such that y = f(x). What is really meant by the statement "y = y(x)" is actually: "we're only interested in those points (x,y), where (x,y) = (x,f(x)) for some function f " (also, somewhat confusingly, labeling the function f, "y" as well).

The important thing to realize, is that when we say "y = y(x)" what we intend to get across is that when we DIFFERENTIATE, we are differentiating with respect to x:

$y' = \frac{dy}{dx}$.

For example, if we want to find the slope of the tangent line to a circle of radius 1 at the point (a,b) on the circle, we do this:

x2 + y2 = 1 is our implicit definition.

Differentiating with respect to x (and using the chain rule), we get:

$2x\cdot \frac{dx}{dx} + 2y\cdot\frac{dy}{dx} = 0$. Note that $\frac{dx}{dx} = 1$.

Solving for dy/dx:

$\frac{dy}{dx} = \frac{-2x}{2y} = -\frac{x}{y}$

Substituting in: x = a, y = b, we get the slope is -a/b. For example, at the point (0,1), the slope of the tangent line is 0, and at the point (√2/2, √2/2) the slope of the tangent line is -1.