Mathematics Stack Exchange
Mathematics Stack Exchange

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 $S\circ R=\{(a,c); (\exists b\in B) (a,b)\in R \land (b,c)\in S\}$.

1202 questions
1
vote
1 answer

Why do composite functions need to include the domain of one of its constitutents

Let $$f(x) = \frac{x}{x-2}$$ Additionally, let $$g(x) = \frac{3}{x}$$ The domain of $f(x)$ is all reals, except for $x = 2$, and the domain for $g(x)$ is all reals, except for $x=0$. $$(f \circ g)(x) = \frac{3}{3-2x}$$ However, the domain for this…
sangstar
  • 1,947
1
vote
1 answer

If $f(g(x))=x$ for all $x$, how few solutions could $g(f(x))=x$ have?

This question is related to this one. Consider a function $g:\mathbb{R}\rightarrow \mathbb{R}$ and its left-inverse $f$ $$\forall x \in \mathbb{R} : f(g(x))=x $$ Then $$g(f(x))=x $$ has a set of solutions (usually a set with the same cardinality as…
Wouter
  • 7,673
1
vote
2 answers

If function g satisfies $g(x)+1 = g(x+1)$, can $g$ be defined arbitrarily for $0\leq x<1$? How?

To try to understand this concretely, I chose function $\mathcal f$ to be defined by $\mathcal f(x) = x + 1$ and the piecewise function: $$g(x)= \begin{cases}\sin(x), & \text{0 $\le x\lt$ 1} \\x-1, & \text{$x\lt$ 0 or $x \ge$ 1}\end{cases}$$ for…
Sourblob
  • 67
1
vote
3 answers

Example of composition of Relations

Question: Let the set $A$ be defined as $A = \{ a, b, c, d \}$, and let the relations $R$ and $S$ on the set $A$ be defined as $R = \{(d, a), (a, b), (b, c)\}$, and $S = \{(a, a), (b, d), (d, c)\}$. Explain why the ordered pair $(a, b)$ is or is…
Joshua Mathews
  • 103
1
vote
1 answer

Suppose $f(x)$ and $g(x)$ are $1$−$1$ functions on their respective domains. Show that $f(g(x))$ is a $1$−$1$ function.

I have an idea of where to go with this proof, but would like a second opinion as to wether I have actually made a logical argument. $f(g(x)) \neq f(g(y))$ where $x, y \in \mathbb{R}$ and $x \neq y$. Given that $g(x)$ is a $1$-$1$ function $g(x)…
Cameron
  • 121
1
vote
3 answers

Composite Function Equality

Let f, g be functions$: \mathbb{R} \mapsto \mathbb{R}$ with $f(x)=x^2+ax+b$ with $a,b \in \mathbb{R}$, such that $(f\circ g)=(g\circ f)$. If the equation $g(x)=x$ has precisely one solution for all $x \in \mathbb{R}$, prove that $(a-1)^2=4b$.
user171110
1
vote
1 answer

Composition of 2 monotonic functions

Let $f$ be a monotonic function $f:[a,b] \rightarrow\mathbb{R}$ and $g$ be a monotonic function $g:[c,d]\rightarrow[a,b]$. Show that $f\circ g$ is monotonic
user233580
  • 11
1
vote
2 answers

Domain and Range of Function Composition

Given: $a\left(x\right)=e^x$ $b\left(x\right)=\left|x+2\right|$ $c\left(x\right)=\frac{\left(x-2\right)}{\left(x+1\right)}$ What is: $\left(\frac{a\cdot b}{c}\right)\left(3\right)$ The domain of $\left(a^{-1}○a^{-1}\right)\left(x\right)$? The…
Jeffrey Liu
  • 21
1
vote
1 answer

Composition of Invertible Functions

Once again we're studying domain and range in class and I encountered this problem. If $f(x)$ and $g(x)$ are both invertible functions, and the domain and range of each function is the set of real numbers, express $\bigl(f\circ g\circ…
user202767
0
votes
2 answers

Composition of Functions from $\mathbb{R}$ to $\mathbb{R}^2$

Is the composition of functions from $\mathbb{R}$ to $\mathbb{R}^2$ a well defined notion? I was asked whether or not the composition of such functions constitutes a binary operation, but I don't know of a standard way to compose such functions!
user82004
0
votes
1 answer

I can't understand this question.

the function f is defined by f(x)=m+x/2+3x for all value of x except when x=h.Find the value of h .
San San
  • 55
0
votes
1 answer

Write the function as a combination of elementary functions

$2xe ^{(-4x^2)}$ Is this correct? $f(x) = -4x^2, g(x) = e^x, h(x) = 2x$ $h(x)\cdot g(f(x))$
user120709
  • 21
0
votes
3 answers

Proofs regarding composition of functions

I'm having trouble approaching the following question: Is the following statement true or false, provide a proof or a counterexample. If $h: A\rightarrow B, \ g: B\rightarrow C, \ f: B\rightarrow C$ are three functions and $ g\circ h=f\circ h $…
user115462
  • 3
0
votes
1 answer

Confusion regarding the definition of composition of functions

This question may seem kinda silly but in constructing a well organized proof about the associativity of function compositions I need to clear my confusion. Here's the definition of composition of relations.See definition of this article from…
Front Left
  • 3
0
votes
0 answers

systems of equations involving composition of functions

If we are given a function $g(x) = x - 1/x$ And another one given in terms of composition $f(g(x)) = x^3 - 1/x^3$ By which general method does one find $f(x)$ ? Can it be found for arbitrary $g(x)$ and $f(g(x)$?
vallev
  • 81
1 2
3
4 5