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

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25 views

Let $f : R → R$. Sequence in image converges, prove convergence of the function at a given value in the sequence.

Let $f : R → R$. Assume $f$ is increasing. Assume $f(1) = 2$. Assume the sequence $2 + (−1)^n/n$ belongs to the image of $f$. Prove that $f$ is continuous at $1$. Should I just show the sequence ...
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0answers
15 views

True/False with regard to uniform continuity on sum of

I am having a lot of trouble proving or disproving these. Let $f_n : E → \Bbb R$ be continuous functions for $1 ≤ n ≤ N$. Let $a_k^{ (n)}$ be $N$ convergent sequences of numbers. Assume $lim_{k→∞} ...
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2answers
51 views

Using the Mean Value Theorem, show that if $f'(x) > 0$ $\forall x \in (a, b)$ then $f$ is increasing on $(a, b)$

Using the Mean Value Theorem, show that if $f'(x) > 0$ $\forall x \in (a, b)$ then $f$ is increasing on $(a, b)$. The Mean Value Theorem states: a function $f$ which is continuous on the closed ...
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3answers
52 views

If $f$ is a increasing function in $[a,b]$, then is it true that $\text{Img}(f) = [f(a),f(b)]$?

If $f$ is a increasing function in $[a,b]$, then is it true that $\text{Img}(f) = [f(a),f(b)]$? I am in doubt because my book said no. Is my proof correct? If $f$ is a increasing function, ...
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2answers
40 views

Prove that if $f$ is a function such that $f'(x) > 0$ for $x \in \mathbb{R}$ then $f$ is a one to one function.

Prove that if $f$ is a function such that $f'(x) > 0$ for $x \in \mathbb{R}$ then $f$ is a one to one function. Set $f(x)$ to be some function such that $f'(x) > 0 \implies$ f(x) is ...
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2answers
65 views

Optimization, find the dimensions of the poster with the smallest area

The top and bottom margins of a poster are 4 cm and the side margins are each 2 cm. If the area of printed material on the poster is fixed at 380 square centimeters, find the dimensions of the ...
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2answers
87 views

Prove whether or not the function is a bijection

Problem statement: Let $ f: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}$ be defined as $ f(m, n) = (3m + 7n, 2m + 5n) $. Is $f$ a bijection, i.e., one- to-one and onto? If yes ...
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1answer
33 views

Operations with functions

I am having issues with a functions problem. I have checked with other friends and they have gotten a different answer than me. I would like if you could check my work for me. I have done this problem ...
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1answer
36 views

Determining Exact Values of Trignometric Equations

Use the special triangles to give exact solutions where possible. Find all values of $x$ such that $0\le x \le 2\pi$ . (a) $\tan^2 x=1$ $\,$ (b)$\, \, 2\cos x + \sqrt{3}=0 \, \,$ (c) $\, \, ...
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2answers
58 views

Analytic functions with infinitely many zeros [closed]

What are some analytic functions, other than the common trigonometric functions, that have infinitely many zeros?
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1answer
9 views

How do I find the value in a range based on the value in another range?

For example Range1 = 0.6 to 0.7 Range2 = 0.0 to 0.5 When Range1 = 0.6(min) Range2 = 0.5(max) Inversely when Range1 = 0.7(max) Range2 = 0.0(min) So lets say the value of Range1 = 0.65, how do I ...
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0answers
40 views

Finding a function using first derivative

I have some data about just first derivative of a function. Also, I know a point of this function(e.g. (x1,y1)). How can I obtain the function? All my date are numerical. dev f(x)=[ 580.00 , 479.7308 ...
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2answers
41 views

Started this problem but can't finish it: Showing pointwise convergence for this summation

I know how to start this problem but am having trouble finishing the end of it. Any help would be great! Thanks We let $g_n: E \rightarrow \mathbb{R}$ be continuous functions for $1 \leq n \leq N$ ...
1
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1answer
51 views

Existence of functions $g$ such that 1. $f\circ g(1) =2$; 2. $g \circ f(1) = 2$, for all $f$ [closed]

Let $S = \{1,2,3,4\}$. Let $F$ be the sets of all functions from $S$ to $S$. a) Prove or disprove the statement: "For all $f \in F$, there exists $g \in F$ so that $(f \circ g)(1) = 2$" b) Prove or ...
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1answer
21 views

Absolute Maximum and Minimum of cos function

I am having a little trouble trying to figure out the following problem: Find the absolute maximum and minimum values of the function $f(x) = x-2\cos x$ on the interval $[0, 2\pi]$. I have taken the ...
0
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0answers
16 views

Find function $f(x,y)$ such that $f(x,0) = J(x)$ and $\nabla_{(x,y)} f(x,y) = g'(y)h(J,\nabla_x J)$?

Let $J:\Omega \to \mathbb{R}$ be a smooth function such that $0 < C_1 \leq J(x) \leq C_2 < \infty$. Is it possible to find a function $f:\Omega \times [0,\infty)$ such that $$f(x,0) = J(x)$$ ...
0
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1answer
44 views

real analysis -functional equation

Be $f:\mathbb R^ +\mapsto\mathbb R$ a function that satisfies the following conditions: a)$ f(f(f(x)))+2x=f(3x)$ for every $x\gt 0$; b) $\lim_{x \to \infty} (f(x)-x)=0$. This was proposed by ...
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1answer
26 views

Quasiconvex functions

Let $\lambda$ be any real number and $f:[0,1]\times[0,1]\rightarrow\mathbb{R}$ given by $$ f(x,y) = \begin{cases} \lambda &\mbox{if } \quad0<x<1, y=1, \\ 1+\lambda y & \mbox{if } ...
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1answer
19 views

Determine if a function is even or odd

Let $f:\mathbb{R}\to\mathbb{R}$. Define $h:\mathbb{R}\to\mathbb{R}$ by $$h(x)=f(x)\{f(x)+f(-x)\}$$ Then, which of the following option(s) is/are correct ? (A) h is even for all f (B) h is odd for ...
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1answer
35 views

How can I get a number between $0$ and $1$, that will be larger as another number is larger?

I have a number $x$, ranging roughly between $0$ and $35000$. I want to create a number $y$, which will be in the range between $0$ and $1$. $y$ will be larger when $x$ is large and smaller when $x$ ...
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1answer
47 views

Solving several equations involving sine function

Recently, I asked for root of this equation $2x - \sin(2x) = \frac{\pi}{2}$, then i got $x = \frac{Dottie}{2} + \frac{\pi}{4}$. Thanks everyone. Now can i define a function like this: $f(n) = x$ to ...
5
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1answer
33 views

Maximum value of function given minimum value

Suppose there is a function $f(x)=\frac{x^2-2x+b}{x^2+2x+b}$ (the problem doesn't specify, but I am assuming $b$ is a real) that has a minimum value of $\frac{1}{2}$. What is the maximum value of ...
3
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2answers
48 views

How does $y=|x+3|+4$ become $y=\frac{1}{2}|2x+3|+4$ (compositions and translations)

Today, I had a test question that was bothering me because my friend and I had different answers to it. It's a grade 12 math question. It's telling us to explain the changes that were made to the ...
1
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1answer
40 views

Logarithm function from graph (word problem)

I was hoping to find some hint to solve this: There is an internet company that wants to price their service $30/month, which will include 25GB on their package. After 25GB, the price will increase ...
2
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2answers
62 views

Is there a non-decreasing function that is discontinuous at every rational point? [duplicate]

A well-known theorem is that if $f:[a,b]\to\mathbb{R}$ is non-decreasing, then $f$ as at most countably many discontinuities. This led me think of the following question. Question: Is there a ...
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1answer
29 views

Is this problem/counterexample stated correctly/valid?

This problem was given by the teacher as a practice exercise. Is it valid? If $f:M \rightarrow N,$ $g:M \rightarrow P$, and $h: P\rightarrow N$ are maps with $g$ surjective and $h$ injective, show ...
0
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1answer
28 views

Find the asymtotes to the function: $f(x)={1+ln|x|\over x(1-ln|x|)}$

Find the asymtotes to the function: $f(x)={1+ln|x|\over x(1-ln|x|)}$. I have difficulties with this one when i try to find the limits, i get indefinites that look like (infinity over infinity)over ...
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1answer
40 views

How do we know that the Maclaurin Series can always be used to approximate a function?

I'm aware of the formula that can be used to derive the Maclaurin series for a particular function: My question is - how do we know that all functions can be represented as an infinite series of ...
3
votes
2answers
92 views

Is $g(x)=\log x$ convex function?

The graph of convex function is : In a book it is written that $g(x)=\log x$ is strictly convex function. So i searched for graph of $g(x)=\log x$ and found that Though it has been said that ...
1
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1answer
32 views

Asymtotes to the the function: $y= \sqrt{{x^3-1\over x}}$ at x=0 there should be a vertical asymtote, but i don't know how to formally execute this..

Asymtotes to the the function: $y= \sqrt{{x^3-1\over x}}$ at $x=0$ there should be a vertical asymtote , but i don't know how to formally execute this. When i try to find the limit, i get the square ...
1
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1answer
27 views

why is the domain of the function in the picture >=0, why is it not just !=0

http://i.imgur.com/Ofmw9ieh.jpg Expression - $$z=\sqrt[131]{\frac{x^2+y^2-2x}{2y-y^2-x^2}}$$ I do not understand why the expression under the odd root has to be greater than or equal to 0 according ...
0
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1answer
49 views

How would you methodically find a tangent/and normal line of a point T$\notin (x,f(x))$? [closed]

How would you methodically find a tangent/and normal line of a point T$\notin (x,f(x))$ ? For example the normal/ and tangent lines from point (-1,-1) to $y=x^2$ . Just interested in 2 dimension ...
0
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1answer
59 views

Simplifying identity cos*cos+sin*cos

$$\cos(3\pi/2 - a) = -\sin(a)$$ According to an answer to one of the questions in my book that's true, but come up with that? $$cos(\pi/2 - a) = \sin(a),$$ but this is $3\pi/2$. do you just ignore ...
2
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2answers
42 views

Showing a complex function is constant

If I know that a function $f$ is entire and $f(z)=f(z+1)=f(z+i)$ for all $z \in \mathbb{C}$, how do I show that $f(z)$ is constant? I feel like this needs use of the uniqueness/identity theorem to ...
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0answers
40 views

Proving existence and uniqueness of solutions to the functional equation $f(n) = r \cdot f(n-1)$

Suppose I have a functional equation $f(n) = r \cdot f(n-1)$ where $r$ is a constant. This represents a geometric progression and a known solution is $g(n) = ar^n$ where $a = g(0)$. By intuition, ...
1
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1answer
28 views

Function inverse

Question: Let $S$ and $T$ be sets and let $f:S\to T$. Show that $f$ is a surjection from $S$ to $T$ iff for each subset $B$ of $T$, $f[f^{-1}[B]]=B$. So it's a surjection if for each element $t$ ...
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0answers
10 views

Choosing a set of functions for linear combinations

I need to approximate some real valuated function $f(x)$ with a linear combination of functions from a set $\{f_i(x)\}$ to solve a practical problem. The domain is rectricted to $x$ in the range ...
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0answers
13 views

Internalizing results about composition and surjectivity/injectivity

I'm trying to see if there is any intuition pump / analogy that allows me to internalize ( and readily derive them) a series of results about the concepts of composition mixed with ...
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2answers
86 views

Showing $\lim_{(x,y) \to (0,0)} \sin (xy) / xy = 1$

How could we show that $$\lim_{(x, y) \rightarrow (0, 0)} \frac{\sin xy}{xy}=1$$ ?? Could you give me some hints ?? EDIT: Could we show it as followed?? $$\lim_{(x,y) \rightarrow (0,0)} ...
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0answers
15 views

Proof of Strictly Increasing Functions

For real numbers c, d with c$<$d we denote the open interval in R by (c,d)=x$\in$R: c$<$x$<$d. Recall that a function f:R-->R is strictly increasing if for all x,y in domain of f, whenever ...
0
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1answer
36 views

How to draw the graph of $x^6 = y^3$ and $3y = (log x)^2$?

For example, if the equation is $x^4 = y^2$ then I can separate this to get the two equations $y = x^2$ and $y = - x^2$ , hence I can plot the two graphs. But how do I simplify the given equation? I ...
2
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0answers
31 views

Functions, sets, intersections

Question: Let $S = \{1,2\} = T$ and define $f: S \to T$ by $f(1) = 1 = f(2)$. Let $B = \{1\}$ and let $C = \{ 2\}$. Find $f[B \cap C]$ and $f[B] \cap f[C]$ and observe they are not equal. Hint: ...
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2answers
44 views

Computation of determinant for Using Inverse Function Theorem

Let $f : \Bbb R^{3} \setminus \{(0, 0, 0)\} → \Bbb R^{3} \setminus \{(0, 0, 0)\}$ be given by $f(x, y, z) = (x/(x^{2} + y^{2} + z^{2}), y/(x^{2} + y^{2} + z^{2}), z/(x^{2} + y^{2} + z^{2}))$. Show ...
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1answer
16 views

For which value of a , b , d these 2 functions are equal

Functions : $$ f(x) = (ax+b)/(x+d) $$ And the inverse one : $$ g(x) = (xd-b)/(a-x) $$ I tried to solve it and I got this : $$ a(-x^2 + ax + b) = d(x^2 + xd -b) $$ But I can't go further, How can I ...
0
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1answer
18 views

Has the functions having countably infinite image, but finite when the domain is bounded, a conventional name?

I'm trying to find properties for functions that cover the following properties and wondering if they have a formal name to search more efficiently. The function $f(x)$ cover the following ...
0
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1answer
37 views

Application of Inverse Function Theorem

This is a seemingly easy exercise. Yet I am not sure if I am missing any finer details here as this is listed as one of the challenging problems on Dr. Epstein's (Upenn) course site for real analysis. ...
2
votes
1answer
33 views

How can we apply the definition?

Show that $$g(x, y)=ye^x+\sin x+(xy)^4$$ is continuous. The definition is: $f : A \mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuous at $x_0 \in A$ iff $\forall \epsilon \exists \delta:$ ...
3
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2answers
57 views

Functions and Sets

Robert, Susan, and Thomas are the sole contestants in a lottery in which two prizes will be awarded. Three tickets with their names on them are placed in a hat. The person whose name is on the first ...
1
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3answers
50 views

Total number of functions $f\colon S\to S$ where $S=\{1,2,3,4\}$

I missed a lecture on this topic and I'm having a hard time figuring out how this discrete function works. I'm given $S=\{1,2,3,4\}$ and $F =$ all functions from $S$ to $S$. What does this mean? I ...
0
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1answer
23 views

What function can produce a perfect saddleback plot and fulfil the following requirement?

I need to find a function that produce a good saddleback plot. The function has the following requirements: Having 2 arguments: x and y Both x and y are natural numbers The result of the function ...