For questions about the composition of functions: 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 $x\mapsto g(f(x))$.

learn more… | top users | synonyms

1
vote
1answer
23 views

Composite Functions

$f(x)= \dfrac{1}{10x+17}+13$ $g(x)= \dfrac{1}{9x-6}$ I need to find $f(g(x)).$ How do I do this? I keep on getting it wrong. The correct answer is $\dfrac{1998x-1202}{153x-92}$. But I am unsure how ...
1
vote
1answer
11 views

Finding inverses of two functions and their compositions to solve for unknown.

$$f(x) = 23x + 27,\;\; g(x) = 12x - d$$ I've found $f^{-1}(x),$ and $\,g^{-1}(x)$, but I don't know how to solve for $d$, given $$f^{-1}(g^{-1}(x)) = g^{-1}(f^{-1}(x)).$$ How do I do this please?
1
vote
2answers
19 views

Which of the following is constant?

If $f,g$ are continuous real valued functions such that $f\circ g$ is constant then which of the following must be constant? $$f,g,g\circ f$$ I think when $f\circ g$ is constant then at least one of ...
0
votes
1answer
17 views

Partial composition

Here is a simplification of my problem. I have the two following functions: $$ f : \mathbb R^{(m+p)} \rightarrow \mathbb R $$ $$ g : \mathbb R^n \rightarrow \mathbb R^m $$ with $m, p,n \in \mathbb ...
0
votes
1answer
12 views

Composing Piecewise Functions

I was wondering how to compose piecewise functions. On a practice exam I was reading, a question asks what F(F(x)) will look like if F(x)= 2x if x<1/2 and = 2-2x if x>=1/2. Would I just ...
0
votes
0answers
11 views

Composition of rational functions degrees

Consider rational function $r(x) = \frac{p(x)}{q(x)}$ with $\operatorname{degree}(p(x)) = n_1, \operatorname{degree}(q(x)) = n_2$ and $gcd(p(x),q(x))=1$. Let degree of $r(x)=n_1+n_2$ (sum of degrees ...
0
votes
1answer
20 views

Mapping of elements notation - Cohn - Classic Algebra Page 13

So Cohn uses the notation that many have wanted to change to, being $xfg$ rather than $g(f(x))$, and I have had the example: Let $f,g: \mathbb{N} \to \mathbb{N}$, be given by $xf = x + 1,xg=x^2$, ...
0
votes
0answers
9 views

Compute the composition of functions

I got wrong in this very odd question for my assignment. Can somebody help me with the answer of this question provided an explantion? Thank you a lot in advance!
0
votes
1answer
27 views

Limit involving the square root and composition

I want to show that if $\lim_{x \to x_0} f(x) = L > 0$, then $\lim_{x \to x_0} \sqrt{f(x)} =\sqrt{L}$. I'm at the point where I have $|\sqrt{x} - \sqrt{x_0}| < \delta$, $\forall \delta < ...
0
votes
0answers
19 views

Prove that the composition of two “closed form functions” is itself a “closed form function”?

So I have been given the definition of a "closed form function" that is a set of functions built inductively (mapping from and to the complex) starting with the fact that the constant functions ...
0
votes
1answer
10 views

Simple composition of linear maps

Let $F : R^2 → R^2$ s.t. $x → (−x_2, x_1)$ and $G: R^2 → R^2$ s.t $x → (x_2, sin x_1)$. Evaluate $G ◦ F$ and $F ◦ G$. I have said $(G ◦ F)(x) = G(F(x))=-x_2,sinx_1$, but I feel as though this is ...
0
votes
1answer
49 views

Finding limits of composition functions of a piecewise?

Having insane amounts of trouble doing this. Heres a graph of $f(x)$: How am I to calculate $\lim_{x \to 0} f(f(x))$ , $\lim_{x \to 3} f(f(x))$ , $\lim_{x \to 0} f(1+x^2)$. One that is even more ...
0
votes
1answer
41 views

How to calculate the limit of f(f(x)) if f(x) is a piecewise function?

Say f(x) is some piece wise function, and on some interval it is a quadratic equation. How do we determine $$\lim_{x \to -2} f(f(x))$$ Would it just be first we sub the quadratic inside the quadratic ...
4
votes
1answer
32 views

Prove that two groups of functions are isomorphic

The two functions $f(x)=\frac{1}{x}$ and $g(x)=\frac{x-1}{x}$ generate, with the operation of function composition, a group $G$ of functions. Prove that this group is isomorphic to the group $S_3$. I ...
8
votes
4answers
1k views

If the absolute value of a function is continuous, is the function continuous?

If $|f(x)|$ is continuous at $a$, is $f(x)$ continuous at $a$? I tried doing it using composite functions. If $g(x)= |x|$, then $g\circ f(x)= |f(x)|$. Since $g(x)$ and $g\circ f(x)$ are continuous, ...
0
votes
1answer
30 views

General method for composition of piecewise defined functions

There is a similarity in questions about composition of functions piecewise defined (see e.g. here, here and here). In these questions the goal is always the same: Given $f,g$ piecewise defined, ...
0
votes
1answer
13 views

How to alter the interval of a composite function

Let $f, g : R → R$ $$f(x) = \begin{cases} x + 3 &\text{if } x ≥ 0,\\ x^2 &\text{if } x < 0 \end{cases}$$ $$g(x) = \begin{cases} 2x + 1 &\text{if } x ≥ 3,\\ x ...
0
votes
2answers
17 views

Manipulation of Composition of Functions

Let $f: A\to B$ and $g,g' : B \to C$. Prove that if $g \circ f=g' \circ f$ and $f$ is surjective then $g=g'$. Is it fair to say that since $f$ is surjective it has a right inverse such that $f\circ ...
0
votes
1answer
58 views

Composite function of $R$-module

Let $M$, $N$, $P$ be $R$-module and functions $f:M\rightarrow N$, $g:M\rightarrow N$, $h:N\rightarrow P$, $k:P\rightarrow M$, which need not be homomorphisms. Define $f+g:M\rightarrow N$ by ...
1
vote
2answers
41 views

What is the 'formula' of a composite function?

Consider $f: \mathbb{R} \to \mathbb{R}$ such that $f(x) = \frac{1}{x^2 +1}$ and $g: \mathbb{R} \to \mathbb{R}\times \mathbb{R}$ given by $g(x) = (3x, x^2)$. I was asked to find the 'formulas' of $f ...
2
votes
1answer
78 views

What's the shape of this “addsTo” function …?

Note that in this combinatronics question, How many lists of 100 numbers (1 to 10 only) add to 700? I was asking: For an array of 100 numbers, each 1 to 10 inclusive, and the total is T - how many ...
1
vote
1answer
28 views

Notation for repeated composition of functions

I have a repeated composition of functions ${T_n}(z) = {\tau _0} \circ {\tau _1} \circ {\tau _2} \circ \cdots \circ {\tau _n}(z)$ By analogy with $\sum\limits_{i = 1}^n {} ,\prod\limits_{i = 1}^n ...
0
votes
1answer
14 views

How do I find and list compositions for (f) and (g)?

Ok, I've literally just spent the last 2 hours just to figure out two compositions problems for homework, and I've about had it. Anyone here that can help? Problem 1 $$ f(x) = 2x(2) - x -3 $$ $$ ...
2
votes
1answer
41 views

Function composition proof

One of the questions assigned to me for homework was: Is it true that $ f\circ (g \circ h ) =f \circ g + f \circ h$? I am in the understanding that $f \circ g$ means $f(g(x))$, so $ f\circ (g ...
0
votes
2answers
46 views

The limit of composition of two functions

I need your help in solving this limits problem. Let $f$ and $g$ be two functions defined everywhere. If $\lim_{u\to b} f(u) = c$ and $\lim_{x\to a} g(x) = b$, then you may believe that ...
0
votes
0answers
20 views

Composition operator of a non-analytic function

I recently came across a problem that can be reformulated in simple terms involving the composition operator $$ C_g\colon X \to X, \quad f \mapsto f \circ g $$ for functions $$ f,g\colon \mathbb{C} ...
4
votes
1answer
46 views

Simplify an iterated function

If we iterate the function $f(x) = \ln(x + 1)$, we get: $$f(f(x)) = f^2(x) = \ln(\ln(x + 1) + 1)$$ $$f(f(f(x))) = f^3(x) = \ln(\ln(\ln(x + 1) + 1) + 1)$$ $$f(f(f(f(x)))) = f^4(x) = \ln(\ln(\ln(\ln(x + ...
0
votes
2answers
23 views

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 ...
0
votes
1answer
46 views

On the existence/applications of infinitely-nested functions

Inside a previous question, one particular nested function shown is the known tetration. This "kind" of arbitrary repeated functions has always intrigued me, because inside their properties lie so ...
1
vote
1answer
28 views

Composition of Ordered Pair

I'm doing math exercises from a Computer Science book and I am confused as to how the following result (from the solutions manual) is obtained: Given the function f={(a,b), (a,c), (c,d), (a,a), ...
1
vote
0answers
35 views

Decomposition of integer polynomials $P(x)$

Let $P(x)$ be an integer polynomial of composite degree $D$. A decomposition is the opposite of a composition. For instance composition of polynomials $A(x),B(x)$ gives $A(B(x)) = C(x)$. The ...
0
votes
2answers
56 views

What does let $F$ denote the set of all functions from $\{1, 2, 3\}$ to $\{1, 2, 3\}$ mean?

Let $F$ denote the set of all functions from $\{1, 2, 3\}$ to $\{1, 2, 3\}$. I'm supposed to prove that this statement is true or false, $$∀f ∈ F, \;∃g ∈ F\tag i$$ so that $g(f(1)) = 2$ But I'm not ...
0
votes
0answers
64 views

Calculus - Proof composition of two function.

$f,g$ are defined in $\mathbb R$ and these conditions apply. $$1) \lim_{x \to x_0} g(x) = y_0$$ $$2) \lim_{y \to y_0} f(y) = L (L\in\mathbb R)$$ $$3)\mbox{ There is a pocked enviroment of }x_0 \mbox{ ...
2
votes
2answers
79 views

Proof for Surjections

I'm reading through Basic Algebra I (which I enjoy so far. Thoughts on this for self-studying?) and am having a difficult time proving surjection. I believe I understand the concept, but when it comes ...
3
votes
2answers
59 views

Composition of functions is constant in $\mathbb{R^2}$.

Let $\hspace{0.05cm}f:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ and $\hspace{0.05cm}$$g:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ be such that$\hspace{0.05cm}$ $g\circ f$$\hspace{0.05cm}$ ...
2
votes
0answers
94 views

Rendering the derivative of composite functions from a graph

I'm on a workbook problem and I want to make sure I'm doing it properly. The problem asks me to find the derivatives of composite functions when given only the graphs of the original functions, here ...
3
votes
2answers
28 views

Composition of convex and power function

Let $g$ be a convex nonegative function, and $p\ge1$. To show: $f(x)=g(x)^p$ is convex. Let $h(x)=x^p$. Then clearly $f=h \circ g$. Denote $\tilde{h}$ as the extended-value extension of $h$, which ...
0
votes
2answers
56 views

If $g \circ f$ is injective, so is $g$

If $g \circ f$ is injective, so is $g$ I don't think this is true. I think that $f$ has to be surjective. So I am going to try to prove that: If $g \circ f$ is injective, and $f$ is ...
0
votes
1answer
37 views

Composition of functions.

If $h(x) = 2x + 1$, find $h(x + 2)$ ...and a second question. Given $f(x)=x^2+1, g(x)=x-2$, then what is the domain of $f(g(x))$ $\begin{align}\text{Work: } f(g(x)) & =(x-2)^2+1 \\ & ...
2
votes
1answer
27 views

Domain of a composite function

I was given the question: Find the domain of the function $f(x)=\ln(\ln(\ln x))$ I found the answer by inspection: $\qquad D(\ln x)=(0,\infty)$ $\therefore\quad D(\ln(\ln x))=(1,\infty)$ ...
0
votes
2answers
45 views

Prove Lipschitz function composed with an integrable function is integrable on [a, b]

Given a Lipschitz function $g$ (i.e. $|g(x) - g(y)| \leq L |x - y|, \forall x, y \in dom(g)$), and an function $f$ integrable on $[a, b]$, how do we prove $g \circ f$ is integrable on $[a, b]$, ...
0
votes
0answers
10 views

Proving IVP with composition equality.

Let $a<b$ for $a,b$ in $\mathbb R$. Let $f,g:[a,b]\to[a,b]$ be two continuous functions such that $f\circ g=g\circ f$. Show that there exists an $x$ in the interval such that $f(x)=g(x).$ I've ...
0
votes
1answer
47 views

Function Decomposition

How do I decompose a function when I'm given $f(g(x))$ and $f(x)$ and the required is $g(x)$? I done some searching on Google and most sites demonstrate the solution where it's left open, they just ...
0
votes
2answers
29 views

Taking the compositions of two constant functions

The questions asks to prove that the composition of g with f is not equal to f with g. However, I don't know whether you can even take the composition of constant functions or how. so if f(x)=2 and ...
1
vote
2answers
78 views

Composition of non-monotonic convex function

Given the following composition of functions: $h:\Bbb R^k\rightarrow\Bbb R$ $g:\Bbb R^n\rightarrow\Bbb R$ $f(x)=h(g_1(x),g_2(x),...,g_k(x))$ There are known rules which guarantee ...
0
votes
1answer
34 views

Need help with compositions of relations

Prove that given relations $R_1 \subseteq A \times B$, $R_2 \subseteq B \times C$, $R_3 \subseteq C \times D$ then $(R1 \circ R2) \circ R3 = R1 \circ (R2 \circ R3)$ I don't know where exactly to ...
1
vote
1answer
25 views

Trig and Inverse Trig Function Compositions

Sorry if this is a dumb question, but I honestly tried searching and all I could find was obvious stuff like $\sin(\arcsin(x)) = x$ So what is the logic behind simplifying expressions like this, ...
1
vote
1answer
24 views

Compostion of functions in taylor series

I was attempting to help answer a question, but I am curious about the following. Suppose we have an analytic function $f(z)\equiv\sum\limits_{k=0}^{\infty} a_k(z-z_0)^k $ on some suitable disk of ...
2
votes
0answers
56 views

Don't know when to add negative numbers

I'm definitely not a math person and only did general mathematics in high school, and unfortunately, not paying as much attention to that as I should have. Well, I'm doing Discrete Mathematics in my ...
0
votes
1answer
57 views

How to solve for $f(2)$ given $3f(x)+f(2-x) = 2x^2$?

I just came across the problem where I'm given that $3f(x)+f(2-x) = 2x^2$ and I need to find $f(2)$. I know it is simple, please help me.