# Questions tagged [function-and-relation-composition]

For questions about the composition of functions and relations.

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### Why $f^2(x) \ne f(x)^2$?

I am working on an exploration which starts with the following affirmation: In this section you studied the Binomial theorem. Recall function composition from earlier in the course. In this context (...
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### Decomposing higher order derivatives of composite functions

I'm new to calculus, and just a little hazy on the skeleton of higher order derivatives when the chain rule is involved. I'm given a table of function and first derivative values, and I need to solve ...
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### How can one find solutions to $f''(x)=f(f(x))$?

This was a differential equation that I came up with. I have never seen any ODEs which involve composition so I have no idea on how to approach this. One solution appears to be $f(x)=0$ but I can't ...
1 vote
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### Example of $f,g:\mathbb{R}\to\mathbb{R}$ such that $f\circ g$ is bijective but $g\circ f$ is not bijective

I am seeking functions $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ such that $f\circ g$ is a bijection and $g\circ f$ is not a bijection. Here is what I have done so far in terms of ...
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### Are the theorems g•f injective/surjective then f/g is injective/surjective invertible?

i have questions. I know that: if $g \circ f$ is surjective than g is surjective; If $g \circ f$ is injective than f is injective; If g and f are both injective than $g \circ f$ is injective; If g and ...
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### Am I using the right formal logical statement to represent the relationship between two sets?

Given two sets named bigger_set andsmaller_set with an element of smaller_set "pointing&...
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### What technique, method or procedure exists for someone to determine a recurring or explicit function that fits a set of data points?

I have an infinite set of data points (the first seven values are posted bellow) and I'm trying to figure out how someone would go about determining the recurring or explicit function from them? ...
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### Composition of relation R = E9 ◦ E12

Given the relations E9 and E12 on the set of integers defined as follows • aE9b precisely when a ≡ b (mod 9). • aE12b precisely when a ≡ b (mod 12). Let R = E9 ◦ E12 (a) Suppose that 2Rx. Prove that x ...
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Consider the discounting condition in the Blackwell's sufficient conditions: (Reference: A related question). There exists some $\beta \in (0, 1)$ such that $[T(f + a)](x) ≤ (T f)(x) + βa$, for all $f ... 0 votes 1 answer 53 views ### Questions Regarding this Chain Rule Proof I saw this proof of the Chain Rule on Hardy's A Course of Pure Mathematics (the notation I use will be a little different). • Chain Rule: Let$f\,\colon Y \subset \mathbb{R} \to \mathbb{R}$,$g\,\...
Let $f^{\circ n}$ denote the $n$-fold composition of function $f$. As an example, $f^{\circ 3}x$ is short-hand for $f(f(fx))$. Is there another form of composition to denote $((ff)f)x$? Would one be “...