Questions tagged [implicit-function]

This tag is for questions relating to "implicit function", a function or relation in which the dependent variable is not isolated on one side of the equation.

The notion of implicit function is of utmost importance while solving real-life problems.

A function in which the dependent variable, and independent variable(s) are not separated (isolated) on opposite sides of the equality are known as implicit function.

i.e., If it only has the form $~f(x,y)=0~$, then it is implicit.

e.g., Take $~x^2 + xy~ – y^2 = 1~$, then $~f(x,y)=x^2 + xy~ – y^2-1=0~$.

Reference:

https://en.wikipedia.org/wiki/Implicit_function

254 questions
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How does this implicit equation work?

I experimented with mathematics and Mathematica and got a very interesting curve, which defined as: $$\sqrt{x^{2}+y^{2}}=\cos\left(\log_{2}\left(\sqrt{x^{2}+y^{2}}\right)\right)$$ After plotting this equation, we get infinite concentric…
PavelDev
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How to find the self-intersection point of $x^x=y^y(x,y>0)$?

As the figure below shows, the graph of the implicit function $$x^x=y^y,(x,y >0)$$ composes of a straight line and an arc, which of the two have an intersection point $P$. How to find the coordinates $(x_p,y_p)$ of $P$? Does there exist a…
mengdie1982
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Limit of a function defined implicitly

Let a function $y:\mathbb{R} \setminus \left\{ 0\right\} \to \mathbb{R}$ be defined implicitly by the equation $F(y(x),x)=0$ for some $F:\mathbb{R} \times ( \mathbb{R} \setminus \left\{ 0\right\} ) \to \mathbb{R}$. Assume that $y$ is well defined,…
Mikhail
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Prove the equation implicitly represents a function

Unfortunately I got stuck solving the following exercise: Prove that the equation $xe^z + ye^{-z} + z = 1$ implicitly defines z as a function of (x,y), in the set $A\times[0,\infty)$ where: $A =…
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How can I show that certain implicit functions can partition $\mathbb R^2$?

Consider the implicit function defined by the equation $$ f(x,y) = 0 \tag{1} \label{eq1} $$ where $f : \mathbb R^2 \rightarrow \mathbb R$ is some continuous function. Suppose that the implicit function defined by \eqref{eq1} partitions $\mathbb R^2$…
mhdadk
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Express y in terms of x for equation $y = 2 x \sin (x) \cos (y)$

The below equation gives beautiful graph in desmos. I was trying to find a way to draw this graph using Javascript but for that I first need to express y in terms of x but I am not able to figure out a way, any pointers would be great. $$y=2x(\sin…
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Showing that an implicit function is continuous

I want to show that the following implicit function is continuous: $$2x = \ln(|x+iy-1|)-\ln(|x+iy+1|)$$ I plottet the function with mathematica and it seems to be continuous, but i would like to show it in an analytic way. Is there a good way to…
putti.123
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Iterative solution of an implicit equation

I have the following equation: $$\ln \left(\frac{\alpha^2}{\beta^2}\right) = \frac{1}{m \xi_\alpha} + \frac{b\ln(m \xi_\alpha)}{m^2}$$ where $\xi_\alpha := \xi(\alpha^2)$. The objective is to solve for $\xi_\alpha$ given that $\ln…
time12
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Solving for $g(k)$ in the implicit equation $(k+1)g(k)+k!=g(k+1)$

Is there any good solution method to find the function $g(k)$ where $k \in \mathbb{N}$ that solves the following implicit equation? $$(k+1)g(k)+k!=g(k+1)$$ I found that using $g(k) = k! - (k-1)!$ evaluates the left side…
Ty Jensen
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Nesting Implicit Functions

If I have 2 equations $y=f(x)$ and $w=g(y)$ then I can nest them to get $w=g(f(x))$. Can a similar thing be done with implicit functions? Suppose I have a 2 equations $F(x, y)=0$ and $G(y, w)=0$. Can I combine them to get an equation relating $x$…
Amaar
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2 answers

Implicit formula using equation in intercepts

First of all I apologize if this is a trivial question, I am not sure if my solution is what the question is asking for. Question :Using an equation in intercepts, obtain an implicit formula f(x,y)=0 for the straight line intersecting the…
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separating the dependent variable in an implicitly expressed function

The following equation is from a force triangle from statics (mechanics) and was written by applying the law of cosines. $F^2=F_R^2+a-bF_R$ (1) where a and b are constants, $F_R$ is the resultant force and $F$ is a component. The question I was…
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Solving implicit equations in R

I have an equation which looks like this: $$\frac{1}{\sqrt{}}=−2\log⁡\left(+\frac{}{\sqrt{}}\right)$$ where $A$ and $B$ are constants How do I solve this in $\mathbb{R}$? Can this be solved iteratively?
ACE
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Implicit function and polynomials

Can someone give me an example what is the difference between an implicit function and curve with polynomials as coordinate system? I am learning basic math concepts and I do not understand those two concepts. So for my homeworks I supplied this…