MHB Implicit Function: Why Is It a Function?

[roni1]

Why the implicit function is a function?

 

[caffeinemachine]

Why the implicit function is a function?

Can you state what you mean by an implicit function?

 

[Olinguito]

An implicit function is a function that is not (sometimes cannot be) given explicitly. The most direct way of specifying a function is to give its value for each member of its domain:
$$f:\{1,2,3\}\to\{1,4,9\};\ f(1)=1,f(2)=4,f(3)=9.$$
It is however more usual to define a function by means of a formula:
$$f:\mathbb Z\to\mathbb Z;\,f(n)=n^2.$$
Formulas can be recursive:
$$f:\mathbb Z^+\to\mathbb Z^+; f(0)=f(1)=1,f(n+2)=f(n+1)+f(n)$$
(which defines the Fibonacci sequence) or somewhat more complicated:
$$f:\mathbb R^+\to\mathbb R;\,f(x)=\int_0^\infty t^{x-1}e^{-t}\,\mathrm dt$$
(which defines the gamma function). These are all explicitly defined functions. But you can also define a function as follows:
$$f:[-1,\,1]\to[0,\,1];\,f(x)\in\{y\in[0,\,1]:y\ge0,\ x^2+y^2=1\}.$$
This defines the upper unit semicircle. In other words, $f$ is the function such that $f(x)\ge0$ satisfies the implicit equation $x^2+f(x)^2-1=0$. This is an example of an implicit function.

Of course the above implicit function can be defined explicitly as
$$f:[-1,\,1]\to[0,\,1];\,f(x)=\sqrt{1-x^2}.$$
But not all implicit functions can be redefined explicitly. One of the most important examples of such a function is the Lambert W function.

If you’re into the rigours of set theory, the formal definition of a function is as follows: Let $A$ and $B$ be sets; then a function $f$ from a set $A$ to $B$, written $f:A\to B$, is the triple $(A,B,G)$ where $G\subseteq A\times B$ such that $\forall a\in A$, $\exists b\in B$ such that $(a,b)\in G$ and $\forall a\in A,\,b_1,b_2\in B$, $(a,b_1),(a,b_2)\in G$ $\implies$ $b_1=b_2$. $A,B,G$ are called the domain, codomain, and graph respectively of the function $f$; if $(a,b)\in G$ we write $f(a)=b$.

 

[HOI]

Why the implicit function is a function?

On the contrary, an "implicit function" is NOT a function but implies a function. A "function" would be of the form y= f(x) where "f(x)" is some formula in the variable x. If, instead, we write 2y+ 8x= 5, we call that an "implicit function" because we can solve for y as a function: y= f(x)= 5/2- 4x. A bit more complicated is x^2+ y^2= 4 which implies two functions, y= f(x)= \sqrt{4- x^2} and y= g(x)= -\sqrt{4- x^2}.