Implicit Function: Why Is It a Function?

In summary, an implicit function is a function that is not always given explicitly, typically defined by a formula.
  • #1
roni1
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Why the implicit function is a function?
 
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  • #2
roni said:
Why the implicit function is a function?
Can you state what you mean by an implicit function?
 
  • #3
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 $f(a)=b$.
 
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  • #4
roni said:
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 [tex]x^2+ y^2= 4[/tex] which implies two functions, [tex]y= f(x)= \sqrt{4- x^2}[/tex] and [tex]y= g(x)= -\sqrt{4- x^2}[/tex].
 

FAQ: Implicit Function: Why Is It a Function?

What is an implicit function?

An implicit function is a mathematical concept where an equation is defined in terms of two variables, and one variable is dependent on the other. This means that one variable cannot be solved for explicitly, but rather the relationship between the two variables can be expressed by the equation.

Why is it called an implicit function?

It is called an implicit function because the relationship between the two variables is implied by the equation, rather than being explicitly stated. This is because one variable cannot be solved for explicitly.

How is an implicit function different from an explicit function?

An explicit function is one where one variable can be solved for explicitly in terms of the other variable. In an implicit function, this is not possible, and the relationship between the two variables is expressed by the equation.

What are some real-life applications of implicit functions?

Implicit functions are used in various fields such as physics, engineering, and economics to model relationships between variables that cannot be explicitly solved for. They are also used in computer graphics and programming to create complex shapes and animations.

How are implicit functions useful in scientific research?

Implicit functions allow scientists to model and study complex relationships between variables without having to explicitly solve for them. This can be useful in understanding and predicting natural phenomena and can also aid in the development of new theories and models.

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