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

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0
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1answer
24 views

What does it mean to 'Find the limit function $f(x) = \lim_{n\to\infty}e^{-n\; x}$?

As the question asks, I'm wondering about what it means to: Find the limit function $$f(x) = \lim_{n\to\infty}e^{-n\;x}$$ By taking the limit, I can easily see that the exponential decay function ...
2
votes
1answer
41 views

How can I compute $\frac{exp(\lambda v_j)}{\sum_{i=1}^n exp(\lambda v_i)}$ in a stable way?

Given an $n$ vector and $\lambda$ > 1e4 I wish to compute this sum $$\frac{exp(\lambda v_j)}{\sum_{i=1}^n exp(\lambda v_i)}$$ for a fixed $j \in \{1, \dots, n\}$. The sum should be less than one, ...
-2
votes
1answer
37 views

Use the Mean Value Theorem to prove an inequality [on hold]

Using MVT, prove the following equation to be true: $$\sqrt{x} - \frac{x-y}{2\sqrt{y}} < \sqrt{y} < \sqrt{x} - \frac{x-y}{2\sqrt{x}},$$ given that $y>x>0$
0
votes
1answer
23 views

Find a one-to-one correspondence (i.e, a bijection) [on hold]

Find a bijection between the following sets where {[]} denotes a closed interval and {()} denotes an open interval A = [-3,7] and B = [41,100] & A = (-∞,-3) and B = (8,∞)
0
votes
1answer
17 views

functions and recursions

The sequence s(k) where k=1,2,3.... satisfies the recursion s(n)=s(n−2)+s(n−3) for n≥4. If s(n) is rewritten in the form s(n)=s(n−1)+S(n0,n1,…) where S(n0,n1,…) is some linear combination of terms ...
0
votes
0answers
55 views

Solve $x+x^{\frac{1}{e^{-x}}}+e^{-x}=p$, where p is some real number

I started to tinker a little bit, discovered that derivative approaches 1 - surprisingly, quite quickly. If f(x) is expression on the left handside, then following inequality applies: \begin{equation} ...
0
votes
1answer
28 views

How can I write the equation of this graph?

I have this simple graph : I want to write this graph in this form : f(t)= {1, if .... and 2, if ...} I can't fill these spaces, I can't come to a solution for this no matter how I try. Can you ...
1
vote
3answers
51 views

Definite integral of function is zero

I am attempting to solve an equation wherein $\int_{-\infty}^\infty f(x) \, dx = 0$. There obviously exist some some solutions, such as $f(x) = xe^{-x^2}$ and trivially $f(x) = 0$, but is there a ...
0
votes
0answers
14 views

behavior of function

Im looking at the asymptotic behavior of the function $f(x)=x-c(\lceil \frac{x}{c} \rceil)$ as $x \rightarrow \infty$ for some constant $c>0$. I believe this function is bounded above by $0$ ...
0
votes
1answer
51 views

Consider $f : \mathbb{N} \to (-1,1) \cap \mathbb{Q}, \ n \overset{f}{\mapsto} \sin(n)$. Is $f$ a surjection? If not can we make it a surjection?

This is not homework! I have recently been thinking about the properties of the sine function and whether it can effectively map elements from certain spaces to entirely "fill out" another space. I ...
-1
votes
0answers
15 views

Natural exponential function Origin question [on hold]

Can the graph of a natural exponential function pass through the origin(0;0) ?
1
vote
1answer
18 views

How to linearize the following?

$$y^2=\frac{π^2\cdot L^2}{3g\cdot x}$$ What should I plot against what to get a linear function (assuming $L$ and $g$ are constants)? I don't know how to tackle this problem.
1
vote
1answer
34 views

Function Transformations

Take the function $f(x)=x^2$. Apply the transformation so that $f(x)=x^2+C$. One option for the two sides of the equation to remain equivalent is for all $y$ values for which $x$ is defined to also ...
0
votes
0answers
18 views

How to write an equation with a denominator parameter less and greater than 1

I am writing a paper, but I do not know what is the most compact and exact formula to express this equation: $$\left| A_y(v) \right|^2 = \hat{q} v^\gamma \quad \text{where} \quad \left\{ ...
2
votes
3answers
48 views

How to define $y= |x+2|+|x-3|$ in a piecewise manner

I need to define the function $y= |x+2|+|x-3|$ over the relevant intervals, but I am not entirely sure what this entails. How do I find the needed intervals? Plugging in different values gives me an ...
0
votes
1answer
72 views

Extend ${\bigl(1+\frac1x\bigr)}^{{x}}$ to $\overline{\mathbb R}$

We can extend these functions to $\overline{\mathbb R}$ by taking limits says here. \begin{align} \mathrm e^{-\infty} &= 0 \\ \mathrm e^{+\infty} &= \infty \\ \ln{\left|0\right|} &= ...
12
votes
5answers
1k views

Are all continuous one one functions differentiable?

I was reading about one one functions and found out that they cannot have maxima or minima except at endpoints of domain. So their derivative , if it exists, must not change it sign , i.e. , the ...
2
votes
0answers
43 views

Prove that $\coth ^2(x)-1\equiv \mathrm{cosech}^2 (x)$

Prove that $\coth ^2(x)-1\equiv \mathrm{cosech}^2 (x)$ My attempts, $\coth^2 (x)-1\equiv(\frac{e^x+e^{-x}}{e^x-e^{-x}})^2-1$ $\equiv \frac{e^{2x}+e^{-2x}+2}{e^{2x}+e^{-2x}-2}-1$ $\equiv ...
-2
votes
1answer
30 views

Calculate the area between functions

[I need to find the area between this three functions, therefore I need to use Integral g(x)-f(x) but I tried and it gives me negative and enormous numbers.]
-2
votes
0answers
16 views

How to solve power series for terms and convergence?

Find the first five non-zero terms of power series representation centered at x=0 for the function. What is the interval of convergence? f(x) = \frac{x}{1+3 x}
0
votes
0answers
11 views

Global extremes and continuity of multivariable function

I am trying to find extremes and continuity of the function $$ g(x,y) = \frac{x}{y}. $$ I have found out that domain is simply $x \in \mathbb{R}$, $y \neq 0$. The derivative should be: $$ ...
0
votes
3answers
32 views

Basic Functions: Adding on to every X [on hold]

Sorry about the title, wasn't exactly sure how to name this. So, I have this table. $$\begin{eqnarray} 1 &=& 15\\ 2 &=& 20\\ 3 &=& 25\\ 4 &=& 30\\ &\ldots\\ 7 ...
0
votes
0answers
25 views

Linear Function Transformations

I don't understand how a transformation can affect just the x variable or y variable. Since isn't y a function of x i.e. f(x) or g(x) etc. I understand that if you start with a function say f(x)=x^2 ...
-2
votes
0answers
27 views

Investigate continuity of $\frac{x-y}{x^3-y}$

Investigate continuity of $$f(x) = \begin{cases} \frac{x-y}{x^3-y}, y\neq x^3\\ 1, y=x^3&\end{cases}$$ How to investigate that? Is it enough to show that when $y=x^3$ then the denominator is zero ...
0
votes
2answers
33 views

Finding Domain$(\cos (\sin x))^{1/2}+\arcsin((1+x^2)/(2x))$

How to find domain and range of following function : $$(\cos (\sin x))^{1/2}+\arcsin((1+x^2)/(2x))\text{ ?}$$
0
votes
2answers
41 views

Range of function $y=(e^x-e^{-x})/(e^x+e^{-x})$

What will be the range of function $y=(e^x-e^{-x})/(e^x+e^{-x})$ ? How should I approach this category of problems with exponential functions? Please Help.
1
vote
2answers
45 views

Construct an explicit bijection $f:[0,1] \to (0,1]$, where $[0,1]$ is the closed interval in $\mathbb R$ and $(0,1]$ is half open.

The problem: Construct an explicit bijection $f:[0,1] \to (0,1]$, where $[0,1]$ is the closed interval in $\mathbb R$ and $(0,1]$ is half open. My Thoughts: I imagine that I am to use the fact that ...
1
vote
1answer
44 views

Do equations that rely on a fractional number of variables exist?

In statistics, data is usually fitted with trend lines. Usually you can get statistics back that say things pertaining to how correlated one variable is with another. For instance if a variable $x$ ...
0
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0answers
7 views

Constructing functions with Given Hessian

Given a matrix-valued function $H(x)\colon R^n\to R^n\times R^n$, what is the condition that there exists a real valued function $f(x)\colon R^n \to R$, such that $H(x)$ is the second derivative of ...
1
vote
0answers
22 views

Determining Equation Based on Number Set [on hold]

I have a set of numbers: 25, 30, 15, 20, 50 for which I'm trying to figure out the function. I'm just wandering if it is possible to find such function and if so, how I could do it.
0
votes
1answer
29 views

The domain of $f(x)=\left(\Bigl\lvert\bigl\lfloor |x|-1\bigr\rfloor\Bigr\rvert-5\right)^{-1/2}$

How should I find the domain of the following function? $$f(x)=\left(\Bigl\lvert\bigl\lfloor |x|-1\bigr\rfloor\Bigr\rvert-5\right)^{-1/2}$$ I am getting something like $6\leq|x|<7$ but not sure ...
2
votes
1answer
23 views

If $F$ is continuous and $\lim_{\left\|x\right\|\to\infty}F(x)=\infty$, then all niveau sets $\left\{x:F(x)\le\alpha\right\}$ are compact

Let $F:\mathbb{R}^n\to\mathbb{R}$ be continuous and $$\lim_{\left\|x\right\|\to\infty}F(x)=\infty\tag{1}$$ I want to show that $$N(F,\alpha):=\left\{x\in\mathbb{R}^n:F(x)\le\alpha\right\}$$ is ...
2
votes
1answer
72 views

If there is an injection $f: X \to Y$ with $m=n$ then $f$ is a bijection.

The Statement of the Problem: Let $X,Y$ be finite sets with $ \lvert X \rvert = m $ and $ \lvert Y \rvert = n $. Prove the following statement by induction on $ m \ge 1$: If there is an injection ...
1
vote
0answers
30 views

What is the reverse image of $f:[0,6\pi] \rightarrow \mathbb{R}$, $f = sinx$?

What is the reverse image of $f:[0,6\pi] \rightarrow \mathbb{R}$, $f(x) = \sin x$? Specifically, I need to find the reverse image of an interval $[a,b]$. I know that this will be a group, as $\sin x$ ...
0
votes
0answers
7 views

Transformation of graphs, finding the values of unknowns

I am a second grade IB student using "Mathematics Standard Level for the IB Diploma, Cambridge" book.This is the question I have a problem with: "Let f(x)=(3x-5):(x-2) a) Find the value of constants ...
0
votes
1answer
28 views

Example of weakly separated function

A function $f:X \rightarrow Z$, where $X$ is a topological space and $Z$ is a metric space, is called weakly separated if for every $\epsilon>0$, there is a neighbourhood assignment $(V_x)_{x \in ...
0
votes
1answer
31 views

Transformation of graphs

Let $f(x)=(3x-5):(x-2)$ a) Find the value of constants $p$ and $q$ such that $f(x)=p+ q:(x-2)$. b) Hence describe a single transformation which transforms the graph of $y=1:x$ to the graph of $y= ...
0
votes
1answer
34 views

Proof by Contradiction to show that if $f^{-1}$ exists, $f$ must be onto

Use proof by contradiction to prove that if $f^{-1}$ exists, then $f$ must be onto where $f:A→B$. Proof: I think the contradiction of the theorem would be: if $f$ exists then $f^{-1}$ must be ...
1
vote
1answer
46 views

Showing $(f^{-1}∘g^{-1})=(g∘f)^{-1}$

If the functions $f$ and $g$ are both bijections then the in inverse of the composition function $(f∘g)$ will exist. Show that it will be $(f^{-1}∘g^{-1})=(g∘f)^{-1}$ For the proof assume ...
0
votes
2answers
21 views

For any prime $p ≠ 2,5$, prove there are at most four values of the last digit of any power $p^{i}$?

I am currently working on this question and I am thoroughly stuck. I believe that this question is saying that for any prime $p$, there will be four or less numerals $p-1$ that exist in the numeral ...
2
votes
2answers
35 views

Give an example of a set $A$ and a function $f\colon A \to A$ where $f$ is onto but not one-to-one.

I am currently trying to decipher this question but I have been unable to thus far. If a set $A$ is mapped onto itself, it seems that you would always have a function that is both onto and one-to-one. ...
-2
votes
0answers
64 views
+100

Find a function so whenever it is near a lattice point $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$

A function $f:\mathbb{R} \rightarrow \mathbb{R}$ is an "easy estimator" if any point on f,$(x_0,y_0)$ is near a lattice point $([x_0],[y_0])$ then $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$. In other ...
0
votes
3answers
38 views

If $\nabla f(x) = 0$ on a set $S$, is $f$ necessarily constant on $S$?

If $\nabla f(x) = 0$ on a set $S$, is $f$ necessarily constant on $S$? I know it is true when $S$ is open convex, or open connected, but what about any arbitrary $S$?
0
votes
1answer
24 views

Limit of $\frac{\sin(xy)}{x}$

Find limit of $$\lim_{(x,y) \rightarrow (1,0)}\frac{\sin(xy)}{x}$$ Is it enough to show that if $(x,y) = \left(\frac{1}{n}, \frac{1}{n} \right)$ then $lim_{(x,y) \rightarrow (1,0)}n ...
2
votes
2answers
30 views

Multivariable function's limit $\frac{y^3-x^3}{x-y}$

Find limit: $\lim_{(x,y)\rightarrow(1,1)}\frac{y^3-x^3}{x-y}$. How to find that. Should i look for two subsequences or what? Do not get this.
0
votes
1answer
12 views

Function that returns an odd number if (n-k)<0 [closed]

I need to find a function f(n, k) which returns an odd number if (n-k) <0 and an even number if (n-k) >= 0.
0
votes
1answer
15 views

what does a critical number is doing for a function??

what does a critical number is doing for a function? Let $f(x)$ be a differentiable function. And $a$ is a critical number. (ie, $f'(a)=0$). Then can you always predict that this point $(a,f(a))$ ...
1
vote
1answer
45 views

N-Tuples or N-functions in category theory

When I'm writing out categories of my Haskell programs, I often get stuck whilst trying to describe morphisms that involve functions that involve more than one argument, such as 2-tuple construction. ...
5
votes
1answer
55 views

Isn't this a non-surjective epimorphism on the category of sets?

I am trying to prove that a morphism in the category of sets is epic iff it is a surjective function. Recall that for objects $A,B,C$, $f \in \hom(A,B)$ is epic when $g_1 \circ f = g_2 \circ f ...
2
votes
0answers
18 views

Oracle for the inverse function

Let $F$ be a 1-1 function from $[0,1]$ onto $[0,1]$, which is continuous and monotonically increasing. Two oracles are given: A direct oracle - given $x\in[0,1]$, it returns $F(x)$. An inverse ...