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

For questions about the composition of functions and relations.

945 questions
Filter by
Sorted by
Tagged with
26 views

### Translation of the function in Complex

Consider the $T : z \to z+a$ for $a \in \mathbb{C}$ (Here the $a = \alpha + i \beta$) Take any complex function, $f(z)$ Then, $T \circ f(z) = f(z) + a$ is translation by $a$ (I.e. $f(z)$ is translated ...
47 views

### Let 𝑎, 𝑏 ∈ ℝ and let 𝑓$_{𝑎,𝑏}$: ℝ → ℝ be the function defined as 𝑓$_{𝑎,𝑏}$ (𝑥) = 𝑎𝑥 + 𝑏 for all 𝑥 ∈ ℝ. Use this for the problems below.

a) Prove that 𝐺 = {𝑓$_{𝑎,𝑏}$ | 𝑎, 𝑏 ∈ ℝ, 𝑎𝑛𝑑 𝑎 ≠ 0} is a group, where the operation is composition. Let 𝐺1 = {$\big(\begin{smallmatrix} a & 0\\ b & 1 \end{smallmatrix}\big)$ ∶ �...
23 views

### Confusion regarding symmetric and anti-symmetric relation

let R be a relation on a collection of sets defined as follows, R { ( A , B ) | A ⊆ B } Here, since it is an improper set, can't we take it as A=B and say that it is both symmetric and anti-symmetric ...
50 views

### How can I prove that a function is invertible, where do I get the function and the necessary data?

"If g is another invertible function, then the compound function $f ∘ g$ is also invertible, and it is fulfilled.." $$( f ∘g) ^{-1} = g^{-1} ∘ f^{-1}$$ I don't even understand where to start....
19 views

### How do I prove or disprove these relation compositions? [closed]

So I am not told if these statements are true or not (although they appear true). I am asked to either prove them in general terms, or disprove them with specific examples. I am confused about ...
16 views

### Composition of a Realtion with identity is the relation itself

I am completely lost as to how to prove the following, although it appears straightforward. I know how this holds for functions, but how do I prove it specifically for relations? I need to provide a ...
29 views

### Continuity possible?

Can functions $f$ and $g$ not be continuous at $8$ while function $f$ of $g$ is continuous at $8$? So far I've tried using piecewise functions to accomplish this but haven't been successful.
17 views

### How to calculate the maximal domain and hence range in this question?

Question: The function $f : R → R, f (x)$ is a polynomial function of degree 4. Part of the graph of $f$ is shown below. The graph of $f$ touches the x-axis at the origin. part a) wants me to find ...
79 views

### Let $f(x) = |x+1|-|x-1|$, find $f \circ f\circ f\circ f … \circ f(x)$ (n times).

Let $f(x) = |x+1|-|x-1|$, find $f \circ f\circ f\circ f ... \circ f(x)$ (n times). I don't know where to start... Should I use mathematical induction? But what should be my hypothesis? Should I ...
23 views

75 views

### Why $f(g(x)) = x$ and $f(x)=g(x)$ imply $f(x)=x$?

I was watching this video from blackpenredpen where he solves the equation $\sqrt{5-x}=5-x^2$ by writing it in terms of "5". However, there's a comment with an alternate solution using ...
69 views

20 views

### Finding the Integral of A Composition of Logarithm, exp, and Trigometric Functions

I have the function $$f(x)=\ln \left(e^{5+6\left(\sum_{k=0}^{100} \frac{\sin \left(k\pi x\right)}k \right)}+e \right)$$ One, is it possible to integrate this without "special functions" (...
60 views

41 views

### Decomposition of a function and chain rule.

This question is about the basic chain rule (and I think of it when I read about calculation of variation in defining distance in manifold using usual Riemannian metrics) and is related to the another ...
58 views

### Can we really compose random variables and probability density functions?

A renowned professor of statistics (whose name I will not reveal here) told me that the notation $p(x)$ makes perfect sense when $p$ is a pdf and $x$ is a RANDOM variable (i.e. a function). I was a ...
26 views

Question: a) Given the functions $f(x) = x + 2$ and $g(x) = 3^x$, determine an equation for (f ∘ g)(x) and (g ∘ f)(x). b) Determine all values for $x$ for which $f(g(x)) = g(f(x))$. *For part a), I ...
34 views

### Composing trigonometric functions

Let $f(x)=\sin(x)$. If $g$ and $h$ are functions on $\mathbb{R}$ such that $g(f(x))= h(f(x))$, can we conclude $g=h$ ? Can we actually compare $g$ and $h$? I am confused. Please, help me.
48 views

52 views

24 views

### Doubt based on composition of functions

I have a question regarding composition of functions, which goes as follows: Let two functions are defined as $$g(x)=\begin{cases} x^2, &-1\leq x<2 \\ x+2, & 2\leq x\leq 3\\ \end{cases}$$ ...
17 views

### Graph of this function

Let f is a real valued function defined from R to R such that f(x)+f(-x)=5 Is this function even , can we plot this function ‘f’ on graph? And what information do we get from this functional equation?...
34 views

I tried to substitute $f(x)$ & $g(x)$ in their places but didn't find a relation; The function beginning bijective or surjective etc have nothing to do with our case I believe $g(f(x))=x^4+2x^3+... 2answers 41 views ### Find linear transformations$U,T:\textbf{F}^{2}\to\textbf{F}^{2}$such that$UT = T_{0}$(the zero transformation), but$TU\neq T_{0}$. Find linear transformations$U,T:\textbf{F}^{2}\to\textbf{F}^{2}$such that$UT = T_{0}$(the zero transformation), but$TU\neq T_{0}$. My solution Let us consider$T(x,y) = (x,0)$and$U(x,y) = (y,...
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: $i)$ ...
Do there exist functions $f,g : R → R$ such that $f (g(x)) = x^2$ and $g( f (x)) = x^3 \text{ , }\forall x ∈ R$. Simply applying $g$ on both sides of equation $1$ and $f$ on equation $2$ ...