Desmos does plot circle when I enter:

x2+y2=1

But when I enter:

f(x,y)=1

f(x,y)=x2+y2

It doesn't plot anything.

This problem does not exist with explicit functions, e.g. when I enter:

f(x)+g(x)

f(x)=x2

g(x)=x3

That is the same as:

x2+x3

Desmos does plot some cubic function.

So I can conveniently plot any explicit function without repetitions of code and typing, but not implicit functions.

Why? Does exist workaround to achieve the same goal/aim/purpose also with implicit functions too?

You can't do that because you're defining f(x,y) twice.

First let's understand the difference between 2 kinds of Desmos statements: Equations and Function Definitions.

These are all equations:

y = 1
y = x
y = x ^ 2 - x + 14
x ^ 2 + y ^ 2 = 1
x + y = 1
x / y = y / x
sin(x) = e^y

They're all relationships between x and y. Desmos can and will always plot equations as long as they only use x and y as variables.

These are function definitions:

f(x) = x
g(x) = x ^ 2 - 2 * x + 1
D(j) = j ^ j
C(y) = e ^ y
h(x, y) = x + y - x * y

They're like equations, but instead of typing y we give the equation a name, e.g. f(x) and then define it. The first 4 functions take one parameter, so they can be transformed to an equation and will be plotted:

f(x) = x                             //y = x
g(x) = x ^ 2 - 2 * x + 1             //y = x ^ 2 - 2 * x + 1
D(j) = j ^ j                         //y = x ^ x
C(y) = e ^ y                         //y = e ^ x

However the last one takes two parameters, so it can't be plotted on the plane.

Now back to the question, try doing this:

f(x, y) = 1
f(x, y) = x ^ 2 + y ^ 2
f(0, 0)

The last line will throw an error:

You've defined 'f' in more than one place. Try picking a different variable, or deleting some of the definitions of 'f'.

And that's exactly the issue, you defined the function f(x, y) in two spots, so Desmos can't tell which one to use to resolve your inquiry.

In the second example, you define f(x) once and g(x) once, and then you tell Desmos to plot the equation f(x) + g(x) which is basically y = f(x) + g(x) = x ^ 2 + x ^ 3.

Another thing is, your f(x, y) function takes two parameters so it'll never be plotted even if it was correctly defined once, because it's 3 dimensional but Desmos is in 2D.

For example:

f(x, y) = 3 x ^ 2 + x * y - 1

This function will be defined successfully without throwing, but won't be plotted because it's 3 dimensional. You can only plot one 2 dimensional 'slice' of it at a time:

g(x) = f(x,0)

This will plot.

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