Is there a name for an equation that takes the following form? $$F(f(x),f^{-1}(x),x)=0$$ A nice example being $$f(x)-f^{-1}(x)=0$$ because the solutions of this equation are their own inverses. WolframAlpha solved this problem, though I had to type the equation as $f(f(x))=x$. Is there any nice method for solving such equations? How do you think WolframAlpha does it? For instance, what if I wanted to solve $$f(x)={f^{-1}(x)}^2+x$$ Which could also be written as $$f(\sqrt{f(x)-x})-x=0$$
2026-03-28 08:08:08.1774685288
Is this a field of study?
176 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in FUNCTIONS
- Functions - confusion regarding properties, as per example in wiki
- Composition of functions - properties
- Finding Range from Domain
- Why is surjectivity defined using $\exists$ rather than $\exists !$
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Lower bound of bounded functions.
- Does there exist any relationship between non-constant $N$-Exhaustible function and differentiability?
- Given a function, prove that it's injective
- Surjective function proof
- How to find image of a function
Related Questions in SOFT-QUESTION
- Reciprocal-totient function, in term of the totient function?
- Ordinals and cardinals in ETCS set axiomatic
- Does approximation usually exclude equality?
- Transition from theory of PDEs to applied analysis and industrial problems and models with PDEs
- Online resources for networking and creating new mathematical collaborations
- Random variables in integrals, how to analyze?
- Could anyone give an **example** that a problem that can be solved by creating a new group?
- How do you prevent being lead astray when you're working on a problem that takes months/years?
- Is it impossible to grasp Multivariable Calculus with poor prerequisite from Single variable calculus?
- A definite integral of a rational function: How can this be transformed from trivial to obvious by a change in viewpoint?
Related Questions in INVERSE
- Inverse of a triangular-by-block $3 \times 3$ matrix
- Proving whether a matrix is invertible
- Proof verification : Assume $A$ is a $n×m$ matrix, and $B$ is $m×n$. Prove that $AB$, an $n×n$ matrix is not invertible, if $n>m$.
- Help with proof or counterexample: $A^3=0 \implies I_n+A$ is invertible
- Show that if $a_1,\ldots,a_n$ are elements of a group then $(a_1\cdots a_n)^{-1} =a_n^{-1} \cdots a_1^{-1}$
- Simplifying $\tan^{-1} {\cot(\frac{-1}4)}$
- Invertible matrix and inverse matrix
- show $f(x)=f^{-1}(x)=x-\ln(e^x-1)$
- Inverse matrix for $M_{kn}=\frac{i^{(k-n)}}{2^n}\sum_{j=0}^{n} (-1)^j \binom{n}{j}(n-2j)^k$
- What is the determinant modulo 2?
Related Questions in FUNCTIONAL-EQUATIONS
- Functional equation $2f\left(\frac{x+y}{2}\right)-f(y)=f''(x)$
- How to solve the integral equation $f(x) = \int_0^x f(x-y)k(x,y)dy+g(x)$ for $f(x)$?
- Easy looking functional equation.
- Constructing a functional equation that has given solution set.
- Solution of $f(ax+b)=kf(x)$ with $k,a,b$ are real numbers
- Deriving $\sin(\pi s)=\pi s\prod_{n=1}^\infty (1-\frac{s^2}{n^2})$ without Hadamard Factorization
- Stationary Condition of Variational Iteration Method
- How to solve the functional equation $f(x + f(x +y ) ) = f(2x) + y$?
- Solution to the functional equation $f(z)=(-1)^zf(1-z)$???
- If $f(a,b)=f(a,c)f(c,b)$ for all $a,b,c$, when can we conclude $f(a,b)=g(a)/g(b)$ for some $g$?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?