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

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1answer
26 views

Riemann sum of $\sin x$ on [0,1]

For each positive integer $n$, define a function $f_n$ on [0,1] as follows: $f_n(x)$= $0$ $\forall$ $x=0$ $\sin$($\pi\over{2n}$) $\forall x\in(0,{1\over{n}}]$ $\sin$($2\pi\over{2n}$) $\forall ...
0
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2answers
19 views

Bijection of a function.

Define the function f: $(2,\infty) -> (-\infty,-1)$ by $f(x)= \frac{-x}{x-2}$. Show that f is bijective. I know i need to prove both injective and surjective, and I was able to solve the equation ...
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1answer
19 views

Let P(x) be a polynomial of degree 4 , having extremum at $x=1,x=2$ and $\lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2$ Then the value of P(2)

Let P(x) be a polynomial of degree 4 , having extremum at $x=1,x=2$ and $\lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2$ Then the value of P(2) is _______ I worked out the limit using L'Hospital got a relation ...
1
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3answers
68 views

Range of f(x) = $\frac{\sqrt3\,\sin x}{2 + \cos x}$ [duplicate]

Can you give any idea about the range of the following function? $$f(x) = \frac{\sqrt{3}\,\sin x}{2 + \cos x}$$
1
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1answer
37 views

Proving Limits of f(x) and f(a+h) are equal

The question asks me to prove that the equality of these two expressions $\lim_{x\to a} f(x)$ and $\lim_{h \to 0}f(a+h)$ provided their limits exist. My answer: Let $x=a+h$ so this $\lim_{h \to ...
1
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1answer
43 views

Proof identity for any function: $F(A) \cap B = F(A \cap F^{-1}(B))$

Let any number $y\in(f(A))\cap B$. We want to show that $y \in f(A \cap f^{-1}(B))$. Then $X \in A$ and $y \in B$. What should I do next?
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0answers
10 views

Integrating along a line of point sources

I have some concentration that radiates from a spherical point, being steadily consumed until it hits zero at some distance $r_{n}$. This is given by $$C(r) = A\left[r^2 + \frac{2r_{n}^3}{r} - ...
1
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1answer
33 views

Prove there exists an element in the function…Beginning function proof

Having trouble with the last part of my proof: Let f: $\mathbb{Z}\rightarrow \mathbb{Z}$ be a function with $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{Z}$. Prove there exists an element $a\in ...
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1answer
17 views

Beginning Proof on functions and Sum of functions

I've been having a hard time with this proof because I do not know where to go from where I am. The proof we were assigned to in class is as follows: Let $f : \mathbb Z \rightarrow \mathbb Z$ be a ...
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2answers
22 views

Find the angle between two tangent lines…

I have such an exercise: Find the measure of angle formed by the tangent lines drawn through $A(2,-1)$ to the following function: $$f:R\to R, f(x)=x^2$$ My solving was going well till I got stuck at ...
2
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0answers
38 views

Prove That $f(n+f(n))=n$

if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function such that $f(1)=1$ and $$f(n)=n-f(f(n-1))\,\,\: \forall n \ge 2$$ Prove That $$f(n+f(n))=n$$
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1answer
34 views

Proof strategy for $(<=)$: If $g \circ f = id_A$, then f onto $\iff$ g 1-1. [Chartrand 3Ed P239 9.72]

For nonempty sets A and B and functions f : A → B and g : B → A, suppose that $g \circ f =$ the identity function on A. $(♦)$ (e) $(<=)$ Assume that $g$ is one-to-one. Because $g$ is a ...
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4answers
60 views

Is $\sin(\arcsin(x))$ equal to $x$?

I have a question. Is $\arcsin(\sin (x))$ or $\sin(\arcsin(x))$ always equal to $x$? And also for all other trigonometric ratios?
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0answers
17 views

Abbreviate this formula if n = 5; number of elements = (1*2)*(2)*(2)*(2)*(2) = 32

I see a binary pattern, but I am not really good at math. I just know that the result for n = 5 would result in 32. In other words: if n = 1; result 2 if n = 2; result = 4 if n = 3; result = 8 if ...
1
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1answer
46 views

What are some practical uses of functions? [closed]

Functions are basically formal equations that relate a set of inputs to output. What are some practical uses for functions and inverse functions?
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0answers
10 views

How to properly clamp Beckmann Distribution

I am trying to implement the Cook-Torrance Microfacet BRDF shading model and I am having some trouble with the Beckmann Distribution: Beckmann Distribution with width parameter ...
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0answers
41 views

A lemma on function spaces

This is a lemma about function spaces. I'm not really understanding it however. Can someone try explaining it to me? Lemma: let $X$ be in |SET| $(Y, d)$ in |MET|, $f_n$, $f$ is in $Y^X$. Then $f_n\to ...
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2answers
32 views

How many functions are there to a set with A elements from a set with B elements?

How many functions are there to a set with A elements from a set with B elements? I'm looking for a short answer for this question. Please help me
2
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4answers
81 views

If $ f(x)=x^2-3x+1$ then $ f(x-2) = ?$

If $ f(x)=x^2-3x+1$ then $ f(x-2) = ?$ I'm not sure how to properly deal with this function and solve for $f(x-2)$.
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2answers
70 views

Notation for Partial Functions

Suppose we have sets $f$, $A$ and $B$ such that $f\subset A\times B$ and $\forall x\in A\space \forall y,z\in B: [(x,y)\in f \land (x,z)\in f \implies y=z]$ i.e. $f$ is a partial function mapping ...
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1answer
41 views

Union of functions

Let $F=\{f(n)\ |\ f:\mathbb N\to\mathbb N\}$ I want to prove that for any $f,g\in F$, there is always an $h\in F$ that is different from $f$ and $g$, and is larger than both of them. I believe that ...
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0answers
23 views

let E=C[X] be a normed space and T∈ L(E)… prove that.. [closed]

Let E=C[X] be a normed space and T∈ L(E). And let $$\||P||_\ = \left\{ \sum ||P^{(n)}||_\infty, \; \; 0 \leq n \leq ∞ \right\}.$$ where $\||P||_\infty$=sup|p(x)|, 0≤x≤1 1- Justify that T:E→E ...
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2answers
35 views

Conditions on the functions $f,g,h,k$ if $f(x)g(y)=h(x)k(y)$

I was working on this problem, and I thought I'd post my answer so people could see if they have a better one: Spivak Calculus, 4th ed., problem 3-18: Suppose $f,\,g,\,h,\,k$ are functions from ...
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0answers
47 views

Let $\,E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. … [closed]

Let $E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. Let us consider the identity $I :X→Y$. Prove that I is continuous and bijective. Calculate $\,||I||$. Prove that $I^{-1}$ is ...
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0answers
36 views

Prove that B is bounded ?? [closed]

Let G be a Banach space and let B be a subset of G. Suppose that f∈G* we have f(B) = {f(x); x∈B} is bounded in R. Prove that B is bounded. Such that G* is the dual space.
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1answer
17 views

Question about plotting indifference curves

So I am doing a last year paper on microeconomics I m asked to find pareto efficiencies , cores and so on within the edgeworth box. So what I wanna know is that whe they give me in this exercise the ...
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1answer
28 views

Find the real parameter $\color{maroon}{a}$ such that the following functions…

How would you solve this exercise? Determine the values of $\color{teal}a$ for which the following functions: $$\color{maroon}{f:(0,\infty)\to \mathbb{R}},\quad \color{violet} {f(x)=\ln x}$$ and ...
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1answer
19 views

Intersection of graphs, and no solution for trig functions.

All I know is the c=asin(x-b) I don't know how to check the values of b for 'no solutions,' in the case of trig functions. Can someone people provide an algebraic method to solve this.
4
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1answer
64 views

Is there a difference between writing $f: X\rightarrow Y$ and writing $f:X\mapsto Y$?

I think I've heard about a year ago that "$\mapsto$" is only used for a bijection, or do they mean the same thing?
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0answers
26 views

How do I find the polynomial degree of f?

Suppose $f$ is an entire function such that $$\dfrac{| f(z) |}{|z|^4} \leq 100|z|^{11}$$ for $z > 120$. Is $f$ a polynomial of degree $\leq15$?
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2answers
29 views

Find real domain of a function results in $x \geq i$

I have an equation of the form $$f(x) = \sqrt{x^3 + x}$$ for which one needs to define the maximal domain, and image and domain are part of $\mathbb{R}$ (real numbers). $$x^3 + x \geq 0 \implies ...
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2answers
79 views

How to prove a function from $\mathbb N\times \mathbb N$ to $\mathbb N$ is bijective. [duplicate]

I am having trouble with this problem: $f\colon \mathbb N\times \mathbb N \rightarrow \mathbb N$ is defined by $f(i,j)=\dfrac{(i+j-1)(i+j-2)}{2}+i$. How do you prove that $f$ is a bijection from ...
1
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1answer
24 views

Definition of upper hemicontinuity of a correspondence.

When using and examining Kakutani's fixed-point theorem, I've got a question about upper hemicontinuity. A correspondence $f:X\rightarrow2^Y$ is a point-to-set mapping. One way to define upper ...
2
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1answer
25 views

Determine the values of real parameters …

If you have an idea, please, do not leave the page, just write it, I will be very thankful. We have the function $$f:R\setminus \{-1 \}\to{R}$$ ...
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2answers
56 views

Real Analysis; injective and surjective functions

Let $f$ and $g$ functions from $\mathbb{R}$ to $\mathbb{R}$ given by $$g(x)= x^2-x$$ and $$f(x)= -\sin x$$ i) Is $g$ injective? ii) is $g$ surjective? iii) is $g$ invertible? iv) is $f$ injective? ...
2
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1answer
36 views

Suppose a function $f : \mathbb{R}\rightarrow \mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x$ belonging to $\mathbb{R}$.

Suppose a function $f : \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x$ belonging to $\mathbb{R}$. Show that (a) $f$ is one-to-one. (b) $f$ cannot be strictly decreasing, and ...
0
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1answer
16 views

Get number of occurences containing a specific number in combinations of N digits?

If I have all the combinations of 3 digits (000 to 999) I want to count how many results contain the digit 4: 456 104 404 ... For 4XX there are 100, for X4X it ...
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2answers
31 views

Proving that function with domain (-1,1) is injective.

Function $g\colon (-1,1) \rightarrow \mathbb R$ is defined by $g(x)=\dfrac{x}{1-x^2}$. Prove that $g(x)$ is injective. Work: I shifted the equation so that it ends up like ...
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2answers
47 views

Proving that a function from $N\times N$ to $N$ is bijective.

I am stuck on this problem: Define $f: N\times N \rightarrow N$ by $f(i,j)=\frac{(i+j-1)(i+j-2)}{2}+i$. How do you prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically ...
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0answers
37 views

Proving that the set of irrational numbers is uncountable [duplicate]

Work: Assume that the set of irrational numbers is countable. Since $Q$ is infinite, it is therefore denumerable. Therefore, there exists a bijective function $f: N \rightarrow Q$. From here I am ...
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0answers
12 views

Domain of function of form $f(x)=\frac{g(x)}{k(x)}$

I just want to know did we have a rule to find the domain of function in form of $f(x)=\frac{g(x)}{k(x)}$ .I know $k(x)\ne 0$ . but in general do we have any rule to compute domain of function like ...
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0answers
21 views

Groups - Compositions

If the f is written to the right of its argument does that mean the composition of $f g$ is actually $g(f(x))$ instead of being $f(g(x))$ which is the notation I'm used to. I ask this because I read ...
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0answers
45 views

Pointwise convergence and uniform

There is a norm $\Vert .\Vert$ in the space $C([a,b],\mathbb{R})$ such that convergence pointwise implies convergence in norm $\Vert .\Vert$ ? I think not because if there would be the natural ...
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1answer
44 views

Proving the chain rule by first principles

I'm currently trying to prove: $(f(g))'(a)=f'(g(a))*g'(a)$ I have been given a proof which manipulates: $f(a+h)=f(a)+f'(a)h+O(h)$ where $O(h)$ is the error function. However, I would like to have a ...
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1answer
40 views

Working out the median of a beta function

I am trying to work out the median of the beta function of $\mathrm{B}(1/2,1/6)$. I have been told the answer to this is $0.9510$ but i'm unsure to get there? Is there a simple formula in order to get ...
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3answers
40 views

inverse of a function f(x)..change x and y

Find the inverse of the function $f(x)= (2x-1)/(x^2-1).$ we switch the x and y letters and then solve the the equation...but it became kind of complicated while solving
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0answers
6 views

Non-monotonic function but Homothetic function

Is it possible for a function to be non-monotonic, but still homothetic? Thank you for your explanations.
1
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1answer
36 views

Proving that $f: N\times N \rightarrow N$ is surjective [duplicate]

I am having trouble proving that the function $$f: N\times N \rightarrow N, \ \ f(i,j)=2^{i-1}(2j-1)$$ is surjective. Work: I know that using the theorem in which $n$ is the product of prime numbers ...
0
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0answers
25 views

Bijective function with different domain and co-domain element count

To be bijective is to be both injective and surjective. Which in other words, have to have a one-on-one match right? Then how am I supposed to come up with a bijective function if the domain has a ...
1
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1answer
38 views

Need helping proving that something is differentiable but not continuously differentiable

I need some help please proving that a function is differentiable at $(0,0)$ but not continuously differentiable at $(0,0)$. The function is as follows... (from $\mathbb{R}^2$ to $\mathbb{R}$) ...