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Questions tagged [functions]

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

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7 views

Relation between conditions when Differential equation is given

for option B , i took h(x)=$f^2(x) + g^2(x)$ and got it to be true. but option A and C not able to check
3
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1answer
81 views

If $f(x)$ Polynomial with real coefficient and $f(0)=1$, $f(2)+f(3)=125$ and $f(x)*f(2x^2)=f(2x^3+x)$ then what is the value of $f(5)$?

If $f(x)$ Polynomial with real coefficient and $f(0)=1$, $f(2)+f(3)=125$ and $f(x)*f(2x^2)=f(2x^3+x)$ then what is the value of $f(5)$? . What I tried $$f(0)=1$$ put x=1 $$f(1)*f(2)=f(3)$$ put x=2 $...
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0answers
18 views
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2answers
22 views

How to prove $f$ is injective $\Longleftrightarrow$ $f(A) \cap f(B)=\emptyset$ for all disjoint $A,B⊂X$

How to prove $f(A) \cap f(B)=\emptyset$ for all $A,B⊂X$ with $A∩B=∅$ $\Longleftrightarrow$ $f$ is injective For "$\Longrightarrow$" Let $x \in A$ and $y\in B$. Since $A∩B=∅$, it is $x\neq y$ and ...
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0answers
5 views

Family of functions with curvature parameter

I am looking for a family of functions to model some data. I found this question, and the functions I need are quite similar: I need to define a family (one parameter) of monotonic curves Just as in ...
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1answer
40 views

Find all function $f:R\to R$ such that : $f(ax)f(by)=f(ax+by)+cxy$ where $a,b,c>0$

If $f :R\to R$ and $a,b,c>0$, then find all function such that : $$f(ax)f(by)=f(ax+by)+cxy, \text{where } a,b,c>0 \text{ for all } x,y\in R$$ My attempt When $x=0$ and $y=0$, we find $f(0)=1$ ...
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1answer
26 views

How to solve the composite function using graph

How to solve the composite function using graph ? I know the analytical method to solve the composite function (i.e to find $f \circ g$ or $g \circ f$). Is there any graphical method to find $f \...
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0answers
15 views

Random recursive function that stays near the initial value

I am working on a procedural CG scene and have populated a starry sky with particles of random size. My goal is to make the stars twinkle - in this case, by varying their size and/or alpha channels. ...
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1answer
12 views

Characterize convex functions on the space of convex bodies

I am interested in convex functions on the space of convex polytopes: Let $\mathbb{R}^n$ denote $n$-dimensional Euclidean space. A convex polytope is the convex hull of a finite subset of $\mathbb{R}^...
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0answers
15 views

At these two second-derivative-like limits of quotients equivalent? [on hold]

I was investigating variations on defining the second derivative of a function $f$ with respect to another function $g$, using the quotient rule. But the quotient rule doesn't hold if the function in ...
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1answer
46 views

How to prove $f^{-1}(f(A))=A \quad \Longrightarrow f(A\cap B)=f(A)\cap B$

$X,Y$ are Quantites and $f:X\rightarrow Y$ a function that is injective. i have already proven that $f^{-1}(f(A))=A$ when $f$ is injective. How to prove $f^{-1}(f(A))=A \quad \Longrightarrow f(A\cap ...
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0answers
17 views

Finding a limit involving F(x) when certain conditions are given

I thought to determine the function first but5 since only one information is given and according to that f(x) has one root alpha and at that point, the derivative has to be zero. So I tried to assume ...
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0answers
31 views

Functions (finding period) [on hold]

The value of $n \in \mathbb{Z}$ for which the function $$f(x) = \frac{\sin nx}{\sin\left(\frac{x}{n}\right)}$$ has period $4\pi$ is A. 2 B. 3 C. 5 D. 4
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How to prove $f(A\backslash B)=f(A)\backslash f(B)$ for every $A,B⊂X$ with $B⊂A$ if $f$ is injective? [duplicate]

How to prove $f(A\backslash B)=f(A)\backslash f(B)$ for every $A,B⊂X$ with $B⊂A$ if $f$ is injective? Unfortunately, I have no base to start with. Hope you could help me out.
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0answers
15 views

$f(A)\cap f(B)=\emptyset$ for every $A,B ⊂ X$ with $A\cap B=\emptyset$ $\Longleftrightarrow$ f is injective. [duplicate]

I have already proven$f(A\cap B)=f(A)\cap f(B)$ if $f$ is injective. But how do i prove $f(A)\cap f(B)=\emptyset$ for every $A,B ⊂ X$ with $A\cap B=\emptyset$ $\Longleftrightarrow$ f is injective.
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2answers
35 views

Finding all continuous function which maps any sequence in geometric progression to another geometric progression

Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for any geometric progression $x_n$ the sequence $f(x_n)$ is also a geometric progression. I tried first by taking constant ...
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3answers
35 views

Return 0 for negative numbers using a short list of functions [duplicate]

I'm using a CAD software that allows you to calculate a value using a small set of built in functions. I have the formula complete for the input (call it 'x') of this formula, I just need to return ...
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1answer
26 views

How to show that the Airy function is Lorentzian

The Airy function used to describe the reflected/transmitted intensity of a Fabry-Perot interferometer has the general form: $$\frac{F\sin^{2}\left(\theta\right)}{1+F\sin^{2}\left(\theta\right)},$$ ...
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1answer
32 views

Why aren't 1 and -1 included in the graph of signum fuction?

source : https://en.wikipedia.org/wiki/Sign_function#/media/File:Signum_function.svg As you can see in the graph that 1 and -1 are open circles and aren't included. But the range of the signum ...
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1answer
45 views

$f: \mathbb{R} \to \mathbb{R},\space\space\space f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^2,\space\space$find $f(3)$ in terms of $f(0)$.

$f: \mathbb{R} \to \mathbb{R},\space\space\space\space f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^2,\space\space\space\space$ Find $f(3)$ in terms of $f(0)$. My approach: $$f(x)-2f(\frac{x}{2})+f(\frac{...
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1answer
22 views

Making a formula that finds the horizontal and vertical distance between two points that change with a new angle.

I am making a Scratch 3.0 game. The shooter sprite is holding a gun slightly off-centre (see images), and I need the bullet to go to the end of the barrel of the gun before travelling forward (as so ...
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1answer
16 views

One to one relation between functions that coincide in certain values

I was reading a proof of multivariable calculus that suggested the following property might be true: Given two functions $f(x)$ and $g(x)$ in $\mathbb{R}^n$, if $\forall x_1,x_2$ $f(x_1)=f(x_2)$ $\...
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0answers
15 views

real analysis - Mean Value/Rolle's Theorem [duplicate]

I'm working on the following: Let $f$ be a differentiable function on $[a,b]$ such that $f'(a) < f'(b)$. Let $c$ be a constant such that $f'(a)<c<f'(b)$. Prove that there must exist a point $...
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1answer
44 views

Given a function $f(x)$ that verifies the following conditions

I have a function that verifies the following conditions: $f(x)$ is even $f(5)=6$ $f(x)$ belongs to $[0,5)$ for $x \in [-2,-1]$ It increases in $(- \infty ,-6)$ $\lim_{x\to 6+ } = + \infty$ The ...
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0answers
11 views

How to prove $f(A\cap B)=f(A)\cap f(B) \quad \text{for every} \quad A,B⊂X$ when $f: X\rightarrow Y$ is injective? [duplicate]

To prove $f(A\cap B)=f(A)\cap f(B) \Longleftrightarrow \text{f is injective}$ Beginning: $f(A\cap B)=\{f(x): x\in (A\cap B)\}=\{f(x): x\in A ∧ x\in B\}$ Is that correct and how can I proceed?
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1answer
24 views

Prove convexity of the given function on $\mathbb{R}^n$

$f:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ is given by $$f(x)=\mbox{max}\{ (z_1-x_1)^{+}, (z_2-x_2)^{+}, ..., (z_3-x_3)^{+} \}$$ where $x=(x_2,x_2,...,x_n) \in \mathbb{R}^n$ and ...
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1answer
63 views

Second derivative of itself

I know the $ \frac{d^2x}{dx^2}= 0 $, since $dx/dx = 1 ...$ But by playing with some equations it is easy to get that $d^2f/dx^2=f''(x)$, so $d^2f=f''(x)dx^2$ and $df=f'(x)dx$, so $df^2=f'(x)^2dx^2$. ...
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1answer
39 views

Question on Function of function polynomial [on hold]

If $f(x)=x^3-12x^2+Ax+B>0$ $f(f(f(3)))=3$, $f(f(f(f(4))))=4$ then what is the value of $f(7)$
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0answers
21 views

Showing equivalences (functions, injective)

I wonder which of these statements are equivalent to each other. X,Y are Quantities. $f: X\rightarrow Y$ is a function. Show the equivalence of the following statements: (i) f injective (ii) $f^{-1}...
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1answer
103 views

Is there a mathematical validity of my claims?

I have a question which is not homework. Actually, I have a hard time asking the question. But I will try to express the question as clearly and clearly as I can. In the question, since I cannot use ...
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1answer
27 views

How to generate function from given graph?

Please provide any formula or step by step guide how to generate function for the following graph. Graph logic: part 1 -> where x <= 3 -> linear relation part 2 -> when x from 3 to 6 -> y ...
4
votes
1answer
55 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 ...
4
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3answers
80 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 "...
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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 ...
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2answers
21 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 ...
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2answers
57 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) $f^{-1}\!\bigl(f(A) \bigr)=A \quad \text{for all}~ A \subset X$
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0answers
30 views

Necessary and sufficient conditions that $\langle \zeta, (ij), \lvert\lvert k\, \ell \rvert\rvert, \xi_M\rangle$ 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: ...
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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 ...
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0answers
48 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 ...
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2answers
46 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
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1answer
27 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
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1answer
39 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 ...
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votes
3answers
31 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
16 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+...
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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
34 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 ...
4
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
vote
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 $...