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Questions tagged [function-and-relation-composition]

For questions about the composition of functions and relations.

For questions about the composition of functions and relations: If $f\colon A\to B$ and $g\colon B\to C$ are functions, their composition $g\circ f\colon A\to C$ is given by $x\mapsto g(f(x))$. Similarly for $R\subseteq A\times B$ and $S\subseteq B\times C$ the composite relation is defined $S\circ R=\{(a,c); (\exists b\in B) (a,b)\in R \land (b,c)\in S\}$.

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Check if $f^i(x), i\in \{1,2,3,4\}$ satisfies closure, commutative properties.

Given $f(x)=\frac{1+x}{1-x},\,x\neq -1,0,1,$ find: (a) Compute the composition $f^2, f^3, f^4.$ On the basis of your findings, what are $f^{10}, f^{100}?$ (b) Let $S$ be the set of distinct functions found in part (a). List the elements of $S,$ show…
jiten
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Composite Function Domain will include infinity or not in the below case

Consider the function h(x) = $f(g(x)) = \frac{3}{\frac{6}{3-x}}$ from $g(x) = \frac{6}{3-x}$ and $f(x) = \frac{3}{x}$ , we will say that h(x) to be not defined at infinity and -infinity as both those points make the denomiator of h(x) to be zero ?…
ProblemDestroyer
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Decompose a function

I have $f(x^2-x)=x$ and I would like to find $f(x)$. Is there a systematic way to do it, which also works for similar composite functions?
Bran
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For a fundraiser, there is a raffle with 50 tickets. One ticket will win a $190

For a fundraiser, there is a raffle with 50 tickets. One ticket will win a $190 prize, and the rest will win nothing. If you have a ticket, what is the expected payoff?
Tyler Redfearn
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$f \circ g = g \circ f$ $\iff$ $f$ and $g$ are two linear functions?

Is $$f(x + y) = f(x) + f(y), g(x + y) = g(x) + g(y) \Leftrightarrow \ g\;o\;f = f\;o\;g $$ true? I have a feeling that it must be true, at least the forward way, but I couldn't come up with a proof.. Any ideas?
Amr Ayman
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Demonstrate the truthfulness of the statement

I have the following statement: Determine if is true that if $g: \mathbb{R} \to \mathbb{R}, f: \mathbb{R} \to \mathbb{R}$ and $ (g\circ f)(x) = x$ therefore $g = f^{-1}$ My attempt was: $i)$ Since $g \circ f$ is injective then $f$ is…
ESCM
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Short and quick question about composition function.

If $f$ is bijective and $g$ is bijective too, is $f(g(x))$ always same as $g(f(x))$. My attempt: I'm trying to look at an example if $f=x+1$ and $g=x+2$. The composition is commutative. But, i don't know in general. If it's not, please show me the…
user516076
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Function composition and commutation exercise

I am learning from a self-paced math course and has this exercise on function composition that I couldn't wrap my head around, can you please help me solve it. I proved that $f^2 \circ f = f \circ f^2$ but I don't know how to answer the why part or…
Muhamad Bhaa Asfour
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Find the composition f(g(x)) when four functions are given

Determine $f \circ g$ for the following functions: $$ f(x) = \begin{cases} -x, & x < 0\\ x+1, & x \ge 0 \end{cases} \quad \text{and} \quad g(x) = \begin{cases} x^2, & x \le 2\\ x+2, & x > 2 \end{cases} $$ I need some help with this exercise. I…
ScoobyDuh
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Composition of functions of given set

If ∀x ∈ {1,2,3,4} , $f$(x) = $ x^2 $ and ∀x ∈ {2,4,3,6} , $g$(x) = $ x+1 $. Find $ (g\; o\; f ), (f\; o\; g)\; and \;Im(f \;o \;g)$ I dont understand the question much. If someone can help me it would be really helpful. $ (g\; o\; f ) (x)$ =…
Shehan Tearz
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Are the following either one-to-one or onto functions?

I just want to see if i'm on the right path in determining if the following are onto or one to one. $f\circ g = 3 \lfloor (x+1)/2 \rfloor$ $g\circ f = \lfloor (3x+1)/2 \rfloor$ Both functions are from set of integers to set of integers. Neither of…
Mandeyo
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Function Composition Formulas

The a's and b's are confusing me! For a), I have deduced that g(x) would be something like $-x+a$ where $a$ = the output of $f(x)$... but then you couldn't write g(x) in terms of x and a... ...so how do I write $h(x)$ from this information? so f(x)…
ChaosNova
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Isolating $f$ from $h(x,y,z)=f(g(x,y,z))$

I know that give the system $h(x)=f(g(x))$, $f$ can be isolated by composijg both sides with $g^{-1}(x)$ resulting in $h(g^{-1}(x))=f(g(g^{-1}(x)))$ which simplifies to $f(x)=h(g^{-1}(x))$. Thats simple enough, but can be the same be done for…
Aaron Quitta
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Function composition where two functions are not equal

Can you find two functions $f$ and $g$ such that they are not equal and $$f \circ g= f,\qquad\text{and}\qquad g \circ f= f$$ where $\circ$ denotes composition of the two functions.
Karan
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Are there certain rules for one-one functions?

When I was doing my exercise about functions, I came across a question that asks to prove that composite function f^-1 g^-1 (x) is equivalent to the composite function (gf)^-1 (x) given random one-one functions for f(x) and g(x). I answered the…
nabu1227
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