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

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0
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0answers
14 views

What is Burl’s monthly payment, and how many payments will he make? [on hold]

Burl uses the TVM Solver to estimate the monthly payment for his mortgage. The TVM Solver screen shows these settings: N=180, I%=5.3, PV=250 000.00, PMT=-2008.70, FV=0.00, P/Y=12.00, C/Y=2.00
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1answer
12 views

Finding a function with a certain behavior

I'm searching what are the keywords or the good links to expand my researches. I would like to get the equation ( for programming purposes ) of more or less this curve: ...
2
votes
2answers
32 views

A question about alternate series involving unit fractions

I don't know exactly how to classify this question. It is not from any homeworks, just something I've been wondering about. Let $A\subseteq\mathbb N$ be a subset that contains at least $n$ elements; ...
0
votes
1answer
15 views

Two variable function - Convex/concave

Consider the function: $f(x,y)=e^{ax+by^{2}}$ I have to find the values for $b$ such that $f(x,y)$ is convex and concave. These are my calculations: $f_{xx}=a^{2}e^{ax+by^{2}}$ ...
2
votes
0answers
34 views

How do I calculate the variation of a function?

I am trying to understand how to calculate the variation of a function. In this regard, the book that I am reading offers the following definition - $$V_g([a,b] = sup \sum_{i=0}^n |f(x_{i+1}) - ...
1
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1answer
29 views

Find the range of the function $f(x) = 4x + 8$ for the given domain $D = \{-5, -1, 0, 6, 10\}$

The question is to find the range of each function for the given domain $f(x)=4x+8$, $D=\{-5, -1, 0,6, 10\}$. Is the range just $R= \{-12,4,8,32,48\}$ or am I mistaken? Could you elaborate why my ...
0
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0answers
13 views

Arithmetic implications of different ways to geometrically construct an Hilbert's curve

I have a question on the relation between the geometric and the arithmetic representation of the Hilbert's space-filling curve. Geometric representation: consider the Hilbert's curve ...
-1
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1answer
60 views

how can I find the convergence of the integral $\int_{0}^{1}\frac{\ln(1+n^2x^2)}{n^2}~dx$ , for $ x \in [0,1]$ [on hold]

I want to check the convergence of the integral $$\int_{0}^{1}\frac{\ln(1+n^2x^2)}{n^2} dx $$ for $ x \in [0,1]$ and n->∞ is a constant so can basically pulled out of the integral but I don't know ...
0
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0answers
21 views

$\log(x)$ as iteration-series: how can this be made correct?

I was tinkering with the question whether the logarithm $\log(x)$ can be expressed by some more useful series than by the Mercator series (in terms of (1+x)) for a certain question. One idea ...
2
votes
2answers
63 views

Find an injective function that maps $\mathbb{R} \to (-\infty, 0]$

I'm looking for any ideas as to a function which maps $\mathbb{R} \to (-\infty, 0]$. I considered $-|x|$ but realised that is not injective.
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0answers
23 views

what is Step function?A general definition of what it is. .And explain about it please. [on hold]

what is Step function?A general definition of what it is. .And explain about it please.
1
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3answers
61 views

how can I find the convergence of the integral $\displaystyle\int_{-1}^{1}\frac{1-x^n}{1-x}$ , $ x \in (-1,1)$ [on hold]

I want to check the convergence of the integral $\displaystyle\int_{-1}^{1}\frac{1-x^n}{1-x}$ , for $ x \in (-1,1)$ but i don't know what to do. Every theory I know it is not working. Can someone ...
0
votes
1answer
22 views

Derivatives relation

Please, help me solve the following problem: Suppose $$\frac{df(x,y)}{dx}>0, \qquad \frac{df(x,y)}{dy}<0, \qquad \frac{d^2f(x,y)}{dxdy}<0$$ Is it true that if $y_{2}>y_{1}>0$ then ...
0
votes
1answer
40 views

Clarification of the topology lemma “Any continuous and open injection of the open disk extends over the circle”

My elementary topology 1 class last semester used the book "Topology: Point-Set and Geometric" by Paul Schick, and covered through the end of chapter 8. I am working through the rest of the book on ...
0
votes
1answer
9 views

What if the input of a simple function question is X?

I know how to answer function questions when they are like: fg(3) when f(x) = x + 3 and g(x) = x^2 But what do I do when the question is like: fg(x) when f(x) = x + 3 and g(x) = x^2 Or for a ...
0
votes
1answer
21 views

Determine a curves position over another curve

if the curve of $y= mx^2 -2mx +m$ is over rhe curve of $y=2x^2 -3$, then the limits of the interval must be my attempt: I dont know which concept i have to use. I only know that discriminant is use ...
4
votes
4answers
70 views

Show that $f(x)=f(y)$ then $|x|=|y|$, where $f(x )=\frac{1+|x|}{x}$

Let $f: \mathbb{R}^{*}\to \mathbb{R}$ function definied by $f(x )=\dfrac{1+|x|}{x}$ Show that $f(x)=f(y)$ then $|x|=|y|$ Indeed, $$f(x)=f(y)\\ \iff \\\dfrac{1+|x|}{x}=\dfrac{1+|y|}{y} \\ \iff \\ ...
4
votes
1answer
116 views

A continuous onto function from $[0,1)$ to $(-1,1)$

How I can construct a continuous onto function from $[0,1)$ to $(-1,1)$ ? I know that such a function exists and also I have a function $\displaystyle f(x)=x^2\sin\frac{1}{1-x}$ which is ...
1
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2answers
28 views

Number of discontinuous values

We have to find the number of values of $x$ at which the function $$ f(x) = \frac{2x^5-8x^2+11}{x^4+4x^3+8x^2+8x+4}$$ is discontinuous. I thought that since both numerator and denominator are ...
0
votes
1answer
14 views

Finding the upper tight bound of a mathematical function. (Big O)

I am trying to understand Big-$O$ notation through a book I have and it is covering Big-$O$ by using functions although I am a bit confused. The book says that $O(g(n))$ where $ g(n)$ is the upper ...
1
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1answer
56 views

Engineering/mathmatics question

I have an equation $M(x)= -15.328x^2+176.44x-352.88$ (a parabola) and also $V(x) = -30.657x + 176.44$. I want to know how to find $x$ where the values of $M$ and $V$ combined are the lowest, I'm ...
1
vote
0answers
22 views

Point which do not lie in domain of $f (\frac{2}{x-2})$

If $$f(x)=\frac{1}{x^2-17x+66}$$ then the points which are not in the domain of $f (\frac{2}{x-2})$ are: $(A) \frac{7}{3}$ $(B) \frac{24}{11}$ $(C) \frac{8}{3}$ $(D) 2$ I have already found that ...
-1
votes
1answer
22 views

Prove that $f^{-1}(Y \setminus B_1) = X \setminus f^{-1}(B_1)$

Let $f:X \to Y$ be a map with $A_1,A_2 \subset X$ and $B_1,B_2 \subset Y$. Prove that $f^{-1}(Y \setminus B_1) = X \setminus f^{-1}(B_1)$ where $f^{-1}(B) = \{x \in X: f(x) \in B\}$. Attempt: ...
5
votes
2answers
399 views

A polynomial of degree 3 that has three real zeros, only one of which is rational.

Find a polynomial of degree 3 that has three real zeros, only one of which is rational. My answer: $(x - \sqrt{2})(x - 3)(x - \pi)$. Is this correct? It does have two irrational zeros, but I'm not ...
-1
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0answers
24 views

Differentiable function f(x)

Let $f(x)$ is a differentiable function satisfying $f'(x) + 100 f(x) ≤ 1 $ Then $f(x) -1/k$ is a non increasing function of $x$ , then we have to find the value of $k $ I tried , but at last ...
1
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0answers
47 views

A curious trigonometric equality

Let's consider the following expression: $(1)\cos(15\sqrt{2}^\circ) = \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}+\frac{i}{2}} +  \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}-\frac{i}{2}}$ The left ...
3
votes
4answers
75 views

Solving a functional equation ( $ f(x-y) = f(x)/f(y)$ )

Consider the functional equation $$f(x-y)=f(x)/f(y)$$ If $f'(0)= p$ and $f'(5)=q$, then what is the value of $f'(-5)$ ? My attempt. Using the equation written above I was able to determine the ...
0
votes
0answers
8 views

Find the maximum value of $f(x,y,z)$ on the interval $x_0<x<g^x(p)$, $y_0<y<g^y(p)$, $0<z<g^z(p)$, $p=p(x,y,z)$

First of all, sorry if I am misusing terms or any tags in the post; I am a bit out of my depths here so I'm just trying to explain things in layman's terms. Now, here's the problem: I am working on ...
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3answers
36 views

Given that an expression $2x^3+px^2-8x+q$ is exactly divisible by $2x^2-7x+6$, determine the value of $p$ and $q$. [on hold]

Given that an expression $2x^3+px^2-8x+q$ is exactly divisible by $2x^2-7x+6$, determine the value of $p$ and $q$.
0
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0answers
10 views

Checking of uniformly continuity of the following functions

Which of the following 4 functions are uniformly continuous? and which are not? I want to know the process/explanation of the solutions.
0
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0answers
29 views

How to prove a function is not positive definite [on hold]

I have a lecture about matrix analysis. I have already know some strategies to prove that the function is positive definite. But I face difficulties when I try to see that the (bounded) function is ...
0
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1answer
33 views

Hi I was wondering if there is any algebraic way to find the zeroes of a cos/sin formula without using the unit circle? [on hold]

I understand how to find the zeroes using the unit circle or just graphing it for any matter for my equation. I was just wondering if there is a formula or an algebraic way I could find them. ...
1
vote
2answers
32 views

Prove that if $g \circ f$ is onto and $g$ is one-to-one, then $f$ is onto

Let $f:A \to B$ and $g:B \to C$ be maps. Prove that if $g \circ f$ is onto and $g$ is one-to-one, then $f$ is onto. Attempt: If $g \circ f$ is onto, then for all $y \in A$, $\exists x$ such ...
4
votes
1answer
18 views

Constructing bijection from set of equivalence classes to another set

Suppose $f:A \to B$ is surjective. Define a relation on $A$ by setting $x\sim y$ if $f(x) = f(y)$. It is clear that $\sim$ is an equivalence relation on $A$. Let $\mathcal{E}$ be the set of ...
1
vote
2answers
40 views

If $ Q(x)= x^2-5x+1 $ , find $ \frac {Q(5+h)-Q(5)}{h} $

If $ Q(x)= x^2-5x+1 $ , find $ \dfrac {Q(5+h)-Q(5)}{h} $ can someone show the steps to reach the answer of $h+5$? I got it down to $\frac{h^2+5h-2}{h}$
2
votes
2answers
19 views

Function that returns 1 when a non whole number, 0 when whole number

The title in this case should be self explanatory. When $x$ has a fractional part greater than $0, y$ should be equal to $1$, and when $x$ is a whole number, $y$ is equal to $0$. Anything that gives ...
2
votes
2answers
11 views

Checking injectivity of a certain function from a union of a family indexed by $K$ to $K$

Let $A = \{ A_k | k \in K \}$ be a family of sets indexed by $K$. By Zermelo's theorem, $K$ can be well-ordered. Now, let $\leq$ be a well-order on $K$. Define $j: \bigcup\limits_{k \in K} A_k \to ...
1
vote
2answers
42 views

Check if $f(x)=2 [x]+\cos x$ is many-one and into or not?

If $f(x)=2 [x]+\cos x$ Then $f:R \to R$ is: $(A)$ One-One and onto $(B)$ One-One and into $(C)$ Many-One and into $(D)$ Many-One and onto $[ .]$ represent floor function (also known as greatest ...
2
votes
2answers
77 views

How many maps $A \overset{f}{\rightarrow} A$ satisfy $f \circ f = f$ with the given set $A=\{a, b, c\}$. A few related questions inside.

I am trying to calculate how many maps $A \overset{f}{\rightarrow} A$ satisfy $f \circ f = f$ with the given set $A=\{a, b, c\}$. I would like to see the explicit mappings and learn how you ...
0
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0answers
15 views

function nondecreasing in both variables, set of discontinuities is a nullset

Let $f\colon [0,1]^2\to\mathbb{R}$ be a function such that $g(x):=f(x,y)$ for any $y$ and $h(y):=f(x,y)$ for any $x$ are nondecreasing functions (the second variable is fixed). Prove that the set of ...
1
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2answers
46 views

Showing there is a constant for which an inequality holds true

I'm supposed to show that for $x>0$ and $p>0$ there is a constant $C$ such that $e^x\ge Cx^p$. The constant $C$ depends on $p$ but not on $x$. After analysing the behaviour of the graphs of ...
0
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3answers
54 views

If $f(x+1)= -f(x-1)$, prove that the function f is a periodic function and its period is $4$

I sure do know that I should arrive to the point where $f(x)= f(x+T)$. I've tried replacing $x+1=a$ but the problem seems to get more and more complicated.
1
vote
3answers
28 views

State the domain of $f^{-1}$

$$f(x)=\sqrt{2x+5}$$ $$x \geq -2.5$$ State the domain of $f^{-1}$ \begin{align} \ x & = \sqrt{2y+5} \\ \ \Rightarrow f^{-1}(x) & = \frac{x^2-5}{2} \\ \end{align} The Mark Scheme says that ...
1
vote
0answers
20 views

Machine Learning: are there other functions similar to the softmax?

Recall in probability and machine learning softmax is defined as: $\sigma(\mathbf{z})_j = \dfrac{e^{z_j}}{\sum_{k=1}^K e^{z_k}}$ for $j = 1, ..., K.$ where $\sigma: \mathbb{R}^k \to (0,1)$ ...
1
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0answers
4 views

Analytical Representation of a Sparse Matrix.

I have a sparse, rectangular array where sparse means that missing entries in the table are non-existant, not zero. The array is "patterned" such that on any one line or column, there is one and only ...
-1
votes
0answers
29 views

limit for a continuous function

If we have a function as follows , then we have to find the value of f(0) so that the function become continuous http://i.stack.imgur.com/oE7TH.jpg For that i tried to find the limit when x ...
0
votes
2answers
23 views

An exhaustion of $C_b(\Omega)$

Consider the space $\Omega=\mathbb{R}^{\mathbb{N}}$ and the space $C_b(\Omega)$ consisting of all bounded continuous functions defined in $\Omega$. Actually we are considering in $C_b(\Omega)$ the ...
0
votes
2answers
39 views

Determining exact value of $\cos (A+B)$ in a specific quadrant

The question reads: Angles $A$ and $B$ are obtuse angles in quadrant 2 (II). If $\csc A = 3$ and $\tan B$ = -1/3, determine the exact value of $\cos (A+B)$. How would I take on this question? ...
0
votes
0answers
29 views

Find All functions couples $ f(x+g(y))=xf(x)-yf(x)+g(x) $ [on hold]

Find All functions couples $f:R \rightarrow R$ , $ g:R \rightarrow R $ $\forall x,y \in \mathbb{R} $ The following equation applies: $ f(x+g(y))=xf(x)-yf(x)+g(x) $
1
vote
3answers
64 views

Solving a trig equation that is quadratic?

I have to solve for $x$ given $$\tan^2 x = 2 + \tan x\;\;\;\;\;\;0≤x≤2\pi$$ I brought it all to one side and set it all equal to zero like: $$\tan^2 x - \tan x - 2 = 0$$ What am i supposed to do ...