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I am confused with the term "implicit and explicit function". In several texts, I read that

a function is explicitly when a variable is set in terms of other

that is, when you have something like $y = 3x$ or $z = 3x-2y + 8$, and so on.

I solved some exercises and see things like $y^2-3x = 6$, and conclude that the function could be $y = \sqrt{3x + 6}$. Then, I can always get the function explicitly setting a variable in terms of another. So, what is an implicit function if always it is possible to set a variable in terms of another?

Could you give some typical examples of when a function is implítica or explicit and why?

Thanks!

Tomi
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4 Answers4

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Not all expressions can be rearranged to function form, i.e. $y=f(x)$. In your example you neglected to write down both solutions to $y^2$. You should have gotten two functions: $y=\sqrt{3x+6}$ and $y=-\sqrt{3x+6}$.

Often we will encounter relationships where when trying to convert to a function if you only taking one root (like you did) will mean you lose half (or more) of it. For example a circle (centered at the origin with radius $r$) is given by $x^2+y^2=r^2$. If you try to rearrange it you get two semi-circles: $y=\sqrt{r^2-x^2}$ and $y=-\sqrt{r^2-x^2}$.

Additionally there are many functions where it is impossible to rearrange it into such a form. Take for example $ye^y=x$. There is no elementary function which can be used to write $y$ as a function of $x$. In some situations like this where it is useful to be able to write it as a function then mathematicians make up new functions. For example $ye^y=x$ can be rewritten as $y=W(x)$ where $W$ is known as the Lambert W-function, however the only way to define this function is by stating that $ye^y=x$.

Here is another example where the relationship is given implicitly and attempting to express it explicitly will break it into three functions: $y^3 – 16x – 14y + 2yx^2 = 0$. It looks like: enter image description here

Ian Miller
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Consider that you are neglecting the negative root, $-\sqrt{3x+6}$ when you set $y$ to be an explicit function of $x$. This is why you need the implicit representation, $y^2-3x=6$ which captures both solutions $y=\pm\sqrt{3x+6}$.

"But why don't you just use $y=\pm\sqrt{3x+6}$?"

Because it not a function, it gives you 2 different y values for each x you input.

Jürgen Sukumaran
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Implicit and explicit are properties of the definition of a function and not of the function itself. You can define the exponential function explicitly by a differential equation and an initial condition:

$$\frac{d}{dx}\exp(x) = \exp(x)\\ \exp(0) = 1$$

or by an explicit equation:

$$\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}.$$

In general, an explicit function definition tells you what the function is. An implicit function definition tells you how to test whether a candidate function is the right function.

Rob Arthan
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I would state it. at this moment, this way:

Function is defined by an explicit expression if one variable is defined in terms of other variables in such a way that we have on one side of the expression the sign of functional dependence and on the other side we have how exactly that dependence looks like, as, for example, is the case in the example $f(x,y,z)=xy+yz+zx$.

Function is defined by an implicit expression if it is not defined by an explicit expression. (This look like cheating, right?)

Now you could ask when it is the case that implicit expressions can be converted into explicit expressions and I really do not know under which general conditions is this true.

To see that it is not always easy (or possible?) to convert implicit into explicit expression suppose that we have expression like this one $\sum_{i=0}^{\infty} f_i(y)x^i=0$, where we let $f_i(y)$ to be, for example, polynomial of degree $i$ in variable $y$.

It does not look easy to write the above defined expression in the form $x=f(y)$ and, I believe it is not possible, at least not in the ways that we are accustomed to.

Farewell
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