Questions tagged [functions]
Elementary questions about functions, notation, properties, and operations such as function composition.
22,049 questions
0
votes
0answers
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
votes
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 $...
-3
votes
0answers
18 views
Counterexample for $f(A\backslash B)=f(A)\backslash f(B)$ if $f$ is not injective. [on hold]
do you have an example? Thanks in advance!!
0
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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 ...
0
votes
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 ...
0
votes
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 $f(0)=1$ ...
1
vote
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 \...
1
vote
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. ...
0
votes
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}^...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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
-1
votes
0answers
23 views
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.
0
votes
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.
4
votes
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 ...
0
votes
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 ...
0
votes
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:
$$\frac{F\sin^{2}\left(\theta\right)}{1+F\sin^{2}\left(\theta\right)},$$
...
0
votes
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 ...
1
vote
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:
$$f(x)-2f(\frac{x}{2})+f(\frac{...
0
votes
1answer
22 views
Making a formula that finds the horizontal and vertical distance between two points that change with a new angle.
I am making a Scratch 3.0 game. The shooter sprite is holding a gun slightly off-centre (see images), and I need the bullet to go to the end of the barrel of the gun before travelling forward (as so ...
0
votes
1answer
16 views
One to one relation between functions that coincide in certain values
I was reading a proof of multivariable calculus that suggested the following property might be true:
Given two functions $f(x)$ and $g(x)$ in $\mathbb{R}^n$, if $\forall x_1,x_2$ $f(x_1)=f(x_2)$ $\...
0
votes
0answers
15 views
real analysis - Mean Value/Rolle's Theorem [duplicate]
I'm working on the following:
Let $f$ be a differentiable function on $[a,b]$ such that $f'(a) < f'(b)$. Let $c$ be a constant such that $f'(a)<c<f'(b)$. Prove that there must exist a point $...
0
votes
1answer
44 views
Given a function $f(x)$ that verifies the following conditions
I have a function that verifies the following conditions:
$f(x)$ is even
$f(5)=6$
$f(x)$ belongs to $[0,5)$ for $x \in [-2,-1]$
It increases in $(- \infty ,-6)$
$\lim_{x\to 6+ } = + \infty$
The ...
0
votes
0answers
11 views
How to prove $f(A\cap B)=f(A)\cap f(B) \quad \text{for every} \quad A,B⊂X$ when $f: X\rightarrow Y$ is injective? [duplicate]
To prove $f(A\cap B)=f(A)\cap f(B) \Longleftrightarrow \text{f is injective}$
Beginning:
$f(A\cap B)=\{f(x): x\in (A\cap B)\}=\{f(x): x\in A ∧ x\in B\}$ Is that correct and how can I proceed?
1
vote
1answer
24 views
Prove convexity of the given function on $\mathbb{R}^n$
$f:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ is given by
$$f(x)=\mbox{max}\{ (z_1-x_1)^{+}, (z_2-x_2)^{+}, ..., (z_3-x_3)^{+} \}$$
where $x=(x_2,x_2,...,x_n) \in \mathbb{R}^n$ and ...
1
vote
1answer
63 views
Second derivative of itself
I know the $ \frac{d^2x}{dx^2}= 0 $, since $dx/dx = 1 ...$ But by playing with some equations it is easy to get that $d^2f/dx^2=f''(x)$, so $d^2f=f''(x)dx^2$ and $df=f'(x)dx$, so $df^2=f'(x)^2dx^2$. ...
-1
votes
1answer
39 views
Question on Function of function polynomial [on hold]
If
$f(x)=x^3-12x^2+Ax+B>0$
$f(f(f(3)))=3$, $f(f(f(f(4))))=4$
then what is the value of $f(7)$
1
vote
0answers
21 views
Showing equivalences (functions, injective)
I wonder which of these statements are equivalent to each other.
X,Y are Quantities. $f: X\rightarrow Y$ is a function. Show the equivalence of the following statements:
(i) f injective
(ii) $f^{-1}...
1
vote
1answer
103 views
Is there a mathematical validity of my claims?
I have a question which is not homework. Actually, I have a hard time asking the question. But I will try to express the question as clearly and clearly as I can.
In the question, since I cannot use ...
0
votes
1answer
27 views
How to generate function from given graph?
Please provide any formula or step by step guide how to generate function for the following graph. Graph logic:
part 1 -> where x <= 3 -> linear relation
part 2 -> when x from 3 to 6 -> y ...
4
votes
1answer
55 views
If $f(x)f(y)+f(xy)\le -\frac{1}{4},\forall x,y\in[0,1)$, show that $f(x)=-\frac{1}{2}$
Let $f:[0,1) \to \mathbb{R}$ be a function such that
$$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
votes
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 "...
-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
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
...
0
votes
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$
0
votes
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:
...
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
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 ...
-1
votes
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
votes
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
vote
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 ...
0
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 ...
0
votes
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+...
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
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 $...
