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Questions tagged [functions]

Elementary questions about functions, notation, properties, and operations such as function composition.

3
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1answer
39 views

If $f(x)f(y)+f(xy)\le -\frac{1}{4},\forall x,y\in[0,1)$, show that $f(x)=-\frac{1}{2}$

Let $f:[0,1) \to \mathbb{R}$ be a function such that $$f(x)f(y)+f(xy)\le -\dfrac{1}{4} \quad \forall\, x,y\in[0,1).$$ Show that $$f(x)=-\dfrac{1}{2} \quad \forall\, x \in[0,1).$$ I have proved that ...
1
vote
1answer
35 views

Partial Derivative Disambiguation

There are at least two substantially different meanings to $\frac{\partial}{\partial x}f(x,\ y,\ z(x))$. The $\partial x$ could mean "with respect to $x$ the independent variable," or it could mean "...
-1
votes
0answers
7 views

Parametric functions

How can I plot a parametric function using Graphing Calculator 3D? I am studing parametric equations and sometimes it would be very useful to plot this equations to help visualization, but I have no ...
3
votes
2answers
20 views

Showing a mapping is bijective if and only if a matrix is invertible

Let $\mathbf{A}$ be an $n\times n$ matrix and let $\mathbf{c}$ and $x_{\star}$ be point in $\mathbb{R}^{n}$. Define the affine mapping $\mathbf{G} : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ by ...
1
vote
2answers
44 views

How to show the equivalence of the following statements? $f^{-1}(f(A))=A$ [on hold]

$X, Y$ are quantities and $f : X → Y$ a function. Show the equivalence of the following statements: (i) $f$ is injective (ii) $f^{-1}(f(A))=A \quad \text{for all A⊂X}$
0
votes
0answers
18 views

Necessary and sufficient conditions that $\langle \zeta, (ij), \lvert\lvert k\, \ell \rvert\rvert, \xi_Y\rangle$ generates $\mathscr{P}_n.$

Throughout I use cycle notation and write maps $m:X\to Y$ on the right of their arguments (e.g. $xm=y$ for $m(x)=y$). Let $\zeta=(12\dots n)$. This question is inspired by the following questions: ...
1
vote
1answer
37 views

Is there an easy expression for multiplicative inverses in $\mathbb Z_p$?

I know that in arbitrary division rings, one can go about finding inverses Euclidean division. But take $\mathbb Z_{11}$ as a simple example. Is there a "nice" expression which yields the inverses in ...
0
votes
0answers
47 views

Finding inverse function for g

A function $g$ is given by $g(x) = x + f(x)$ , where $f:[0,1]\rightarrow [0,1]$ and $g:[0,1]\rightarrow[0,2]$ and $f(\sum_{n=1}^{\infty}\frac{a_n}{3^n}) = \sum_{n=1}^{\infty}\frac{b_n}{2^n}$ . How to ...
-1
votes
2answers
39 views

Number of functions $f$ on $\{1,\cdots,7\}$ s.t. $f(f(x))$ is constant

Let $A = \{1,2,3,4,5,6,7\}$. Find the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Please help me out. I tried all sort of combinations but not reaching ...
0
votes
1answer
26 views

Useful bijections

Could someone please provide me with some useful bijections one ought to know for an upcoming examination on cardinality with an emphasis on proofs? For example, the bijective mapping $f : (-1, 1) \...
1
vote
1answer
36 views

Term for “inverse image” of element under set-valued map?

Say I have a function $f: A \to 2^B$. Given an element $b \in B$, I want to refer to the set $f_b := \{a \in A: b \in f(a)\}$. Is there a standard name for such sets? Notice it's not technically ...
0
votes
3answers
28 views

Prove that no function $f : Z → {1, …, 100} $ is one-to-one.

This seems obvious to me, but I'm not sure how I would prove it. Is simply proving $|Z| > |{1...100}| $ sufficient? If so, how would I go about proving that? I know Cantor's theorem that says some ...
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0answers
15 views

Fundamental theorem of algebra and quaternions

I'm not sure if the fundamental theorem of algebra extends to every possible and imaginable numbers (real, complex, quaternions, etc.) but here's my question anyway. Let $f(x) = x^2-2ax+(a^2+b^2+c^2+...
0
votes
2answers
31 views

Let $f:A\rightarrow B$ be a function, and $X\subseteq A$. Prove or disprove that $f(f^{-1}(f(X)))=f(X)$.

Let $f:A\rightarrow B$ be a function, and $X\subseteq A$. Prove or disprove that $f(f^{-1}(f(X)))=f(X)$. Let $A=\mathbb{N}$, $B=\mathbb{R}$ and $X=\mathbb{N\setminus\left\{0\right\}}.$ Hence, $f$ is ...
1
vote
2answers
32 views

Show that the image of the function $f:(0,\infty)\rightarrow \mathbb{R}$, $f(x)=x+\dfrac{1}{x}$ is the interval $[2,\infty)$. [duplicate]

Show that the image of the function $f:(0,\infty)\rightarrow \mathbb{R}$, $f(x)=x+\dfrac{1}{x}$ is the interval $[2,\infty)$. If $x=1$, then $f(1)=2$. So how can I show that the mage of the function ...
3
votes
2answers
140 views

Order between one to one functions and their inverses

Let $f,g :R \to R $ one to one functions such that $f(x)< g(x), \forall x \in R $ Is it true that $f^{-1}(x)>g^{-1}(x), \forall x \in R$?? I'd say yes, thinking at their graphs: if the graph ...
1
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4answers
49 views

Functions that have the same derivative

Let’s say I have two continuous functions $f(x)$ and $g(x)$ , and both have the same derivative $h(x)$. How could I formally show that $f(x)=g(x)+c$ where $c$ is a constant. I know I have to show that ...
2
votes
2answers
22 views

Proving an unspecified function is both one-to-one and onto

Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$ is continuous. Moreover, suppose for every $(x, y) \in \mathbb{R} \times \mathbb{R}$, we have $$|f(x) - f(y)| \geq c|x - y|$$ for some positive ...
1
vote
3answers
37 views

Show the function $f(n)=\dfrac {(-1)^n (2n-1)+1} {4}$ is a surjection

Let $f:\mathbb{N}\rightarrow\mathbb{Z}$ be a function which $f(n)=\dfrac {(-1)^n (2n-1)+1} {4}$. Show that $f$ is surjection. Proof. Let $m\in\mathbb{Z}$. We need to find $n\in\mathbb{N}$ such that $...
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votes
0answers
30 views

Derivation of correlation coefficient using bell curve [on hold]

You can use python to add up all those distances between the data point and the line of best fit, given by the fact that -x/a is perpindicular to ax. I plugged the sum of all the distances in to the ...
0
votes
0answers
21 views

Distance between two functions must be greater than 1

The functions used for this problem are simplified functions: I have a function $g(x)=x_1^2$ and I have a function $h(x,b)=x+b$ and the Area (let's say in the interval x=[0,5]) between these two ...
1
vote
2answers
184 views

Is continuity necessary to establish $f(x+r) =f(x) +f(r), r\in\Bbb Q \Rightarrow f(x+y) =f(x) +f(y) $

Let $f:\mathbb{R} \to \mathbb{R} $ be a function such that $f(x+r) =f(x) +f(r) $,$\forall x\in \mathbb{R} $ and $\forall r \in \mathbb{Q} $. I know that if $f$ were continuous, then we would have $f(x+...
0
votes
5answers
37 views

What's wrong with my proof that $f(x) = x^2, x^4, \ldots$ are bijective?

Let $n$ be an odd positive integer. Prove $f : \mathbb{R} \rightarrow > \mathbb{R}$ defined by $f(x) = x^n$ is bijective. My attempt: First we prove injectivity. Suppose we have $f(x) = f(y)$ so ...
0
votes
0answers
27 views

Explain few terms related to functions [on hold]

What is a $1.$ Increasing Function $2.$ Continuous Function $3.$ Increasing and continuous Functions $4.$ Increasing and discontinuous Functions Explain all this things using graph and give atleast ...
1
vote
2answers
36 views

Range of $\frac{x}{1+x}$

What is the range of $\frac{x}{1+x}$ My approach :- $$y=\frac{x}{1+x}$$ $$y=\frac{x}{x(\frac{1}{x}+1)}$$ $$y=\frac{1}{\frac{1}{x}+1}$$ $y$ is min. If $\frac{1}{x}$ is max and $\frac{1}{x}$ is max ...
0
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0answers
26 views

about cardinal ,mapping on X [duplicate]

if A is a infinite set ,define B={f|f is 1-1 correspondence on A} I want to compute Card(B) My attempt when A is finite ,then card(B)=$2^{card(A)}-2$ Such as A={$a_1,a_2$},then there exist two ...
2
votes
1answer
13 views

Correct algorithm for finding regions of increasing / decreasing and the relationship to critical points.

Yet again, I find myself confused about something that seems basic: critical points and regions of increasing / decreasing. Previously, I thought that to identify regions of increasing / decreasing, ...
3
votes
3answers
86 views

Bijection from $\Bbb N\to \Bbb Z \times \{1, 2, 3, 4\}$.

Find a bijection from $\Bbb N\to \Bbb Z \times \{1, 2, 3, 4\}$. Ok, so I know some element $x$, in $\Bbb N$ maps to an element $(y,z)$ in $\Bbb Z.$ I know to to get from $x$ to $y.$ But since $z$ ...
0
votes
0answers
16 views

Largest number k one-to-one function

What is the largest number k for which there exist a one-to-one function f : {x ∈ Z : x < $k^3$} → {1,3,5,7,9,11} So if k is an integer, and x is in Z, wouldn't there be an infinite amount of x &...
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0answers
25 views

Why can't these two mappings be bijective?

Let $\phi : \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a continuously differentiable function and define the mapping $\mathbf{F} : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ by $$\mathbf{F}(x, y) = (\...
1
vote
3answers
27 views

Question on the injectivity of a function.

Consider the map $f\colon\mathbb{S}^1\to\mathbb{S}^1$ defined as $f(z)=z^2$, where $\mathbb{S}^1$ is the unit circle, $$\mathbb{S}^1=\{z\in\mathbb{C}:|z|=1\}.$$ On $\mathbb{S}^1$ we define the ...
0
votes
2answers
27 views

Absolute value on both sides equation

I have a question: Solve $ 2 - |x+1| = |4x-3| $ The answers are 4/5, 2/3. I understand why one of the answers is 4/5 but what I dont understand is why its 2/3. I created three regions for the ...
1
vote
1answer
45 views

Solving the following linear algebra question [on hold]

Let $f(x)=\mbox{ max }\vert x_i\vert$ for $x=(x_1,x_2,\ldots,x_n)^t\in\mathbb{R}^n$ and let $A$ be $n\times n$ matrix such that $f(Ax)=f(x)$ for all $x\in\mathbb{R}^n$. Prove that there exists a ...
0
votes
2answers
53 views

Smooth approximation of $f(x)=\begin{cases}0&\text{if}\;x<0\\x&\text{if}\;x \geq 0 \end{cases}$ [on hold]

I'd like to find a smooth function to approximate $$f(x)=\begin{cases}0&\text{if}\;x<0\\x&\text{if}\;x \geq 0 \end{cases}$$ This function should be differentiable everywhere. Thanks.
0
votes
1answer
16 views

Solution of functional equation with sums and products

I'm not very familiar with functional equations,so I really need help. Is there some method to obtain functions $b_i, i=1,2$ from $$\displaystyle{\Big(a_1(u)b_1(v)+a_2(u)b_2(v)-\sqrt{a_1^2(u)+a_2^2(u)}...
0
votes
1answer
21 views

Solving equations that involve the modulus

Im often stuck on these types of equations, usually because I don't know which solutions to select as the intersection points. E.g. this question: $ 2|x|= 3 + 2x -x^2 $ To solve this the method I ...
1
vote
0answers
33 views

Existence proof for bijection

Consider two sets $A, B$ and two bijective maps $f_1, f_2$ defined between them, such that $$ f_1: A \rightarrow B, \text{ s.t. } h(a)\leq h(f_1(a)) \text{ for all } a \in A \\ f_2: B \rightarrow A, \...
3
votes
3answers
77 views

How does one prove such an equation?

The problem occurred to me while I was trying to solve a problem in planimetry using analytic geometry. for $b$ between $-\frac{1}2$ and $1$ : $\sqrt{2+\sqrt{3-3b^2}+b} = \sqrt{2-2b}+ \sqrt{2-\sqrt{...
0
votes
4answers
37 views

Minimum value of $y=\sin( 2x) - x$, where $x\in [-\frac{\pi}2,\frac{\pi}2]$

I tried applying the concept that at minima, derivative of $y$ with respect to $x$ should be zero, but realised that it fails as the domain is restricted. Rightly, upon plotting the graph, we can see ...
0
votes
2answers
86 views

Why does $(x \to a) ≠ (x = a)$ , But $f(x \to a) = f(x = a)$ [on hold]

I am really satisfied that $(x \to a) ≠(x=a)$ and if that is not right , Then all the process of $Limits$ is dividing by zero and that is a crime. Since $(x \to a) + h = (x=a)$ , $h ≠ 0$,So Why does $...
0
votes
1answer
12 views

Horizontally translating surge functions [on hold]

is there any way to horizontally translate a surge function (y=Axe^-bx)? TIA
0
votes
1answer
12 views

Find a reduce math function with increase param value

I have a d value which is a range from $0\to5$. I want to build a math function with $d$ param so that return a value from $0$ to $1$. if $d$ increase, function decrease faster. Example $f (d = 0) = 1,...
0
votes
0answers
54 views

Check my math - Number of one-to-one functions $f$ from $\{1, \ldots, n\}$ to $\{1, \ldots, 2n-1\}$ such that $f(x) \neq 2x - 1$ for all $x$

What is the number of one-to-one functions $f$ from the set $\{1, 2, \ldots, n\}$ to the set $\{1, 2, \ldots, 2n − 1\}$ so that $f(x) \neq 2x − 1$ for all $x$? I'm not sure if I did the question ...
0
votes
1answer
22 views

Sketching graphs of functions with a repeated root in the denominator

$$f(x)= \frac{x}{(1-x)^2}$$ I have been trying to sketch functions with a repeated root in the denominator. However, I cannot do it as I struggle to find where $x$ intersects the graph and the shape ...
0
votes
2answers
49 views

Notation to express $(n^1+n^2+…+n^k)$.

What are some mathematical conventions for expressing $(n^1+n^2+...+n^k)$ in a simpler format?
-1
votes
2answers
36 views

Range of $f(x)$ when $e^x + e^{f(x)} = e$. [on hold]

How to find range of the function $f(x)$ in the equation below : $$ e^x + e^{f(x)} = e $$
2
votes
3answers
41 views

How to create a function that returns original inputs from generated output, with no prior knowledge of inputs?

I'm looking for a way to convert 4 numbers into 1 number, then convert that 1 number back into the original inputs with no knowledge of said inputs. 4 Numbers Convert into 1 number (summing/...
0
votes
0answers
19 views

Find the range of the following equation

Find the range of function $$f: [0,1]\to\mathbb{R}, f(x) = x^3-x^2+4x+2\sin^{-1}x$$ I like to know calculus solution if possible.
0
votes
1answer
19 views

What kind of function satisfy those conditions?

I need a function to compute score of object, and the function should satisfy following conditions: takes two variable x and y, output z; small x, small y output z1; small x, big y output z2; big x, ...
0
votes
1answer
13 views

Define Mode function in non continuous region. [on hold]

Let $\Bbb f : R \to R$ be a function defined as $f(x)=|x|/2x$ $\forall x \in R$ {0} Can $f(0)$ be defined in a way such that $f$ is continuous at 0?