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

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2
votes
0answers
26 views

Monotonic function non-continuous in each rational [duplicate]

How can I prove that exists a monotonic non-decreasing function $f: [0,1] \rightarrow \mathbb R$ that isn't continuous in every rational of its domain?
12
votes
8answers
8k views

How do I prove that a function is well defined?

How do you in general prove that a function is well-defined? $$f:X\to Y:x\mapsto f(x)$$ I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the ...
1
vote
2answers
79 views

Finding the domain of an inverse function

I have a clarification to make with my notes. This is about functions. It defines the function $$f(x) = \sqrt[5]{x+1} \space\text{from} \space x=(-1,\infty) \space \text{to} \space[0,\infty)$$ And ...
0
votes
2answers
95 views

Domain of definition of $\frac{x}{\sin(x)}$.

Would the domain of definition of $\displaystyle\frac{x}{\sin(x)}$ be $0$? The function on a graphing calculator looks like it has a lot of infinite limits, so I'm not sure.
0
votes
4answers
419 views

How do we find the period of the function, $x(t)=1+\cos(2\pi t)$

What is the period of the function, $x(t)=1+\cos(2\pi t)$ when, $t=0$, $x(0)=2$ when, $t=\frac{1}{2}$, $x(\frac{1}{2})=0$ when, $t=1$, $x(1)=2$ when, $t=-1$, $x(-1)=2$ when, $t=-\frac{1}{2}$, ...
1
vote
2answers
227 views

Given $f:\mathbb R \to \mathbb R$, $f$ is continuous, $f(f(x))=x$. Find $f(x)$.

Given $f:\mathbb R \to \mathbb R$ $f$ is continuous $f(f(x))=x$ Find $f(x)$. I only find $f(x)=x$ and $f(x)=a-x$. Are there other solutions? Thank you.
2
votes
2answers
109 views

How to make a formula as X value increases from 0 to infinity, Y starts from 1 and approach a ceiling of 2?

The expected curve looks like below. ^                    |2____________________  ...
1
vote
1answer
396 views

What does colon mean between function inputs

I would like to calculate a function, but I am a bit confused with F(x,y:U,s) notation, $$g\left( {x,y:\theta ,f} \right) = \exp \left\{ { - \frac{1}{2}\left[ {\frac{{x_\theta ^2}}{{\sigma _x^2}} + ...
83
votes
6answers
3k views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?

I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success: Find a continuous ...
1
vote
2answers
105 views

If $f\colon B \to C$ is a bijective function and $g\colon A \to B$ is injective, what can we say about $fg\colon A \to C$?

I think $fg\colon A \to C$ is bijective as well but I am not sure. What do you guys think? Any help would be really appreciated. Thanks!
1
vote
1answer
75 views

If a polynomial $P(x)$ tends to $0$ when $x\to\infty$ then $P(x)=0$.

How could I formalize that, when a polinomial $P(x)=a_0+a_1 x +\ldots + a_n x^n$ satisfies $$ \lim_{x\to\infty}P(x) = 0 $$ then $P(x)=0$ for all $x\in\mathbb R$? I've tried to show that supposing that ...
0
votes
1answer
110 views

Extend the domain of a function

I get back to a question I post long time ago, because that is quite important to me... Let $\mathbb{X} = \{a, b, c...\}$ be a finite set, $\mathbb{N}$ refers to the set of all natural numbers. I ...
3
votes
5answers
675 views

What is the difference between $\arg\max$ and $\max$?

What is the exact difference between $\arg\max$ and $\max$ of a function? Is it right to say the following? $\arg\max f(x)$ is nothing but the value of $x$ for which the value of the function is ...
-1
votes
1answer
140 views

Find total number of mappings [duplicate]

Let $A$ be a set of $n$ elements and $B$ be a set of $m$ elements. Show that the total number of mappings from $A$ to $B$ is $m^n$.
9
votes
2answers
145 views

What is the name of this property of a function

I'm trying to find the right vocab word to describe a concept: In computational geometry, there's a concept of a polygon "monotone" with respect to a line. Which means that the polygon intersects ...
1
vote
1answer
124 views

Difference of 2 notations with powerset

Let $\mathbb{N}, \mathbb{V}$ two sets, $\mathcal{P}(\ldots)$ means the power set of a set. $\mathcal{P}({\mathbb{N}})\rightarrow \mathbb{V}$ can be the type of a function mapping a part of ...
2
votes
1answer
65 views

Define a domain filter of a function

Let $\mathbb{B}, \mathbb{V}$ two sets. I have defined a function $f: \mathbb{B} \rightarrow \mathbb{V}$. $\mathcal{P}(\mathbb{B})$ means the power set of $\mathbb{B}$, I am looking for a function ...
5
votes
2answers
159 views

Function notation terminology

Given the function $f:X\longrightarrow Y$, $X$ is called the domain while $Y$ is called the codomain. But what do you call $f(x)=x^2$ in this context, where $x\in X$? That is to say - what is the name ...
1
vote
0answers
35 views

Relations between complex functions satisfying a specific condition

What is the relation between the following two complex functions: $$g(\theta)=\sum_n x[n]\ y[n]\ e^{in\theta}$$ and $$f(\theta)=\sum_n \left(x[n]\pm i\sqrt{1-x[n]^2}\right)\ y[n]\ e^{in\theta}$$ ...
-1
votes
2answers
101 views

Prove that a function on $A\setminus B$ is a function on $A$ minus a function on $B$ if $f$ is injective

Let $f: X\rightarrow Y$ be a map and let $A,B$ be subsets of $X$. Prove that $$f(A\setminus B) =f(A)\setminus f(B) \iff f(A\setminus B)\cap(B)=\emptyset.$$ Deduce that if $f$ is injective, then ...
2
votes
2answers
94 views

questions on a continuous, injective, surjective

Let $f: X\rightarrow Y$ be a continuous, injective, surjective. Question 1, if $f$ is open or closed, then does $f^{-1}$ continuous? Question 2, if $f$ is open or closed, then does $f^{-1}$ open or ...
0
votes
1answer
62 views

A question on $C_p(X)$

What is the sufficient condition on $X$ which makes the spce $C_p(X)$ is the first countable? Thanks!
0
votes
0answers
34 views

Expressing functions using Karnaugh map [duplicate]

Using the Karnaugh map, express the following function: $F(0, 1, 4, 5, 8, 10, 11, 12, 13, 15)$ would this be the answer I'm a little confuse ($b_1=0$ and $b_0=0$) or ($b_3=0$ and $b_1=0$) or ...
0
votes
1answer
89 views

Taking limkits of tricky functions

Hi can anyone help me with this limit. 1) $\sqrt{5-\left(\frac{1}{\sqrt{1+\frac{y^2}{2}}}\right)}$ as $y\rightarrow -\infty $ I am struggling to do the first one, if it can be done using software ...
1
vote
1answer
44 views

Name a stable output of a function taking 2 arguments

$\mathbb{C}$ is a fixed finite set, a fair chaotic sequence $(c_n \in \mathbb{C})$ is defined such that $\forall c \in \mathbb{C}, \exists n_0 \in \mathbb{N}, n > n_0 \wedge c_n = c$. That means ...
1
vote
2answers
282 views

Dirac delta function

1)Prove that the dirac delta function property: $$ x\delta'(x)=-\delta(x)$$ 2)and : $$\int_{-\infty}^\infty \delta'(x)f(x)dx=-f'(0) \ $$
1
vote
3answers
43 views

Correct notation of function with given property

I require a function with the following property: $$ f(x) = \begin{cases} x & x \ge 0 \\ 0 & x \lt 0 \end{cases} $$ This function will be used within an integral, e.g. $$ \int_0^T ...
2
votes
4answers
177 views

How do I “convert” a hyperbolic function into a parabolic function?

How can I find a parabolic function that mimics a hyperbolic one? How would I find the parabolic function for the hyperbolic function $y=5\cosh(\frac x5)$?
2
votes
2answers
71 views

Notation of an iterated function on 2 sets

Let $X$ and $C$ be two sets, I have defined an iterated function on them $f: X \times C \rightarrow X$. What interests me is the iterations of $f$ on an initial value $x \in X$, and a sequence ...
4
votes
1answer
94 views

Determining the bigger of two numbers : $\left(\frac12\right)^e$ or $\left(\frac1e\right)^2$

The question says - use the function $f(x)=sin(x)^{sin(x)}$, where $0<x<\pi$, to determine the bigger of two numbers: $\left(\frac12\right)^e$ or $\left(\frac{1}{e}\right)^2$. Any tips on how to ...
0
votes
3answers
49 views

How to deal with finding pattern in functions

Let $f$ be a polynomial such that $\forall x$; $f(x^2 + 2) = x^4 + 10x^2 + 4$ How would I go about finding a pattern for this? I know that $(x^2 + 2) ^2 = x^4 + 4x^2 + 4$ but that doesn't help much ...
2
votes
4answers
124 views

Algebraically Solve Limit

$$\lim_{x \to 0} \dfrac{2\sqrt{x+1}-x-2}{x^2}$$ I can solve it using l'Hôpital but just cannot find a way to do it algebraically.
0
votes
3answers
522 views

Inverse of sum of two functions

Assuming two functions are invertible, is it true that the inverse of the sum of the two functions is the sum of the inverses (assuming all functions are well behaved)?
4
votes
2answers
46 views

Is my logic correct?

The question says there is a function $f(x)$ which maps $R$ to $R$, and $f''(x)>0$ for all x. This means $f'(x)$ is always increasing. And it is given that ...
2
votes
2answers
5k views

How to find critical points of an absolute values function

I am asked to find How many critical points does the function $g(x) = |x^2 − 4|$ have? I know that the result is $3$ but I can only find $2$. What I do, is to equal the equation to $0$, so $x^2-4=0$ ...
1
vote
2answers
1k views

Let $f$ and $g$ be differentiable functions with the following properties: [closed]

$g(x)>0$ for all $x$ $f(0)=1$ If $h(x)=f(x)g(x)$ and $h'(x)=f(x)g'(x)$, then $f(x)=$ a) $f'(x)$ b) $g(x)$ c) $e^x$ d) $0$ e) 1 Please give me an explanation as well. Thank you very much.
0
votes
2answers
292 views

Describe whether the function is one to one and onto. If both, describe its inverse.

$f:R^{2}\rightarrow R^{2}$ where $f(x,y)=(2x+y,x+y)$. So I know that the function is one to one and onto and don't need help proving that. But with its inverse, I am a little confused. My professor ...
0
votes
3answers
613 views

Reading a 3D graph

How do you read this graph: Function is: $f(x, z) = 1-e^{xz}$ How would I read the graph say $x = 3, z = 0.8$, which would make $y = 0.9092820647$ How do I find $y$ on the graph? Thanks.
4
votes
2answers
98 views

If $f$ = $f^{-1}$ then $f(x) = x$ for some $x$

I would like to know if the following suffices to prove the proposition below. While I can't see anything wrong with it, it gives me a strange feeling. Proposition: If $f$ is a continuous function ...
2
votes
2answers
82 views

Another about limits, vertical asymptote

I am asked to find the vertical asymptotes if any of the following rational function: $$\begin{align} y= (x^2-1)/(x^2-x)\end{align}$$ so what I do is first of all is to find the domain of the ...
4
votes
4answers
227 views

Functions of algebra that deal with real number

If the function $f$ satisfies the equation $f(x+y)=f(x)+f(y)$ for every pair of real numbers $x$ and $y$, what are the possible values of $f(0)$? A.  Any real number B.  Any ...
0
votes
4answers
47 views

Algebra that includes functions and graphing

The answer to the following is B. Can someone explain me how it is please?
0
votes
0answers
60 views

The parent function of this particular graph

I'm looking for the rough "parent function" of the following graph. The graph has a vertical asymptote and a horizontal asymptote. It is most similar to f(x) = ln(x), but more "compressed." Anyone ...
1
vote
1answer
155 views

Is this function convex or concave on $(x,y,z)$?

Is this function convex or concave on $(x,y,z)$? $A$, $B$, $a$, $b$, and $c$ are positive constants. $$f(x,y,z) = A\exp\left(\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}\right) + ...
-4
votes
2answers
402 views

Please help to find function for given inputs and outputs [closed]

Can you help with finding the formula for these input and output values? When $n=1$: $f\left(1,1\right)= 0.0000000000$ When $n=2$: $f\left(1,2\right)= 0.0000000000$ $f\left(2,2\right)= ...
0
votes
0answers
149 views

From point-wise to essential supremum of a set of real-valued measurable functions

I want to prove some property about essential suprema and I think I can show them for the pointwise supremum $\sup S$. The problem is, that the sets involved are uncountable and thus, the point-wise ...
4
votes
0answers
137 views

Terminology Question: Precompose vs Compose?

I was wondering if there was a standard convention on what 'precompose' means compared to 'compose', as I am often confused between the two when all sorts of text casually use both terminologies. For ...
0
votes
3answers
91 views

How to show that $\frac{z-a}{a-z}$ has not inverse?

How to show that $\frac{z-a}{a-z}$ has not inverse? I know that $$\frac{z-a}{a-z}=\frac{-1(a-z)}{a-z}=-1 .$$ But state if I'm wrong, that following is true: $$f(z)=\frac{z-a}{a-z} \Leftrightarrow ...
0
votes
2answers
194 views

Addition function injective?

I am just curious to know if addition of two numbers an injective function? Lets say $\operatorname{Sum}(a,b) = a + b$ Now is the $\operatorname{Sum}$ function an injective functions?
1
vote
1answer
38 views

The shape of the functions

How many of the following functions on R are increasing on their domain? $y = e^x$, $y = x^2$, $y = x^3$ (a) 0 (b) 1 (c) 3 (d) 2 How many of the following functions on R are concave up on their ...