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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
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
$\displaystyle y^2 + \sin y -\sqrt{y}=10$
the above line just means
$\displaystyle (f(x))^2 + \sin (f(x)) - \sqrt{f(x)} = 10$
and probably more importantly
$\displaystyle y'(x) = \frac{dy}{dx}=\frac{df}{dx}=f'(x)$
Is it any clearer now?
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
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:
$\displaystyle y' = \frac{dy}{dx}$.
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:
$\displaystyle 2x\cdot \frac{dx}{dx} + 2y\cdot\frac{dy}{dx} = 0$. Note that $\displaystyle \frac{dx}{dx} = 1$.
Solving for dy/dx:
$\displaystyle \frac{dy}{dx} = \frac{-2x}{2y} = -\frac{x}{y}$
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.
