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

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0
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0answers
4 views

Horizontal stretch factor: what is the correct way?

While at my summer school, I was taught that to translate a function, we should first isolate the x. Suppose the translated function is given by: $$ y = ...
1
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0answers
28 views

How do we address a function whose values are again functions?

One may call a function whose values are functions simply a function-valued function. But is there a canonical name for such an object? A $k$-form on some open $A \subset \mathbb{R}^{n}$ is an ...
0
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1answer
27 views

Derivative and and function terminology

In mathematical parlance, we say "take the derivative of a function f" to indicate that we are computing a new function, which maps slopes, that derives from f. However, in physics, we say "take the ...
1
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8answers
75 views

What is a surjective function?

I am a 9th grader self-studying about set theory and functions. I understood most basic concepts, but I didn't understand what is a surjective function. I have understood what is an injective ...
0
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1answer
19 views

Given a functional equation, its nature, value of its differential at a critical point, what are some methods to calculate its integral over a period?

My particular question is : If $f$ be a decreasing continuous function satisfying $$f(x+y)=f(x)+f(y)-f(x)f(y)$$$$ \forall x,y \in \mathbb{R}; f'(0)=-1$$ then $$\int_0^1f(x)dx =?$$ Answer to this ...
3
votes
0answers
16 views

Domain values of inverse funtion

If I'm plotting $$y=3e^{{x\over3}+1}$$ from $x=0$ to $x=1$ and on the same axes I want to plot its inverse $$y=3\ln\left({x\over3}\right)-3$$ but only for the domain values of $x$ given by the range ...
4
votes
1answer
21 views

Vector fields and tangent vector fields?

I am wondering if there are times when people would call a tangent vector field simply by a vector field? Are not these two concepts different? For example, a vector field assigns (say) to each ...
2
votes
4answers
107 views

Inequality $\left(\sum\limits_{k=0}^{2n-1}\frac{x^k}{k!}\right)\left(\sum\limits_{k=0}^{2n-1}\frac{(-x)^k}{k!}\right)\leq1$ for all $x\in\mathbb R$

I am trying to find a proof of the following inequality $\left(\sum\limits_{k=0}^{2n-1}\frac{x^k}{k!}\right)\left(\sum\limits_{k=0}^{2n-1}\frac{(-x)^k}{k!}\right)\leq1$ for all $x\in\mathbb R$ and ...
-5
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0answers
30 views

Matrices and Linear Algebra- Determine if the list is linearly independent in the real vector space. [on hold]

1.Determine if the list $((3,2,0,1),\,(2,1,4,0),\,(0,-1,12,-2))$ is linearly independent in the real vector space $\mathbb R^4$. 2.In the real vector space $C(\mathbb R,\mathbb R)$ of all continuous ...
0
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1answer
12 views

show multivarable functions are one-to-one, onto.

$F:\mathbb{R}^3 \rightarrow \mathbb{R}^3, F(x,y,z)=(2x,y,3z+y)$ My current method for these sort of questions is to try to find the matrix that represents this transformation and then see if i can ...
0
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0answers
19 views

With 2 as smallest period of the function $f(x)$= $\tan^2[(\frac{\pi x}{n^2-5n+8})]$ + $\cot(n+m)\pi x$ ;the period m can't belong to is?

Here n $ \in N$ , m $\in Q$. Options are: A) $(-\infty, -2) \cup (-1, \infty)$ B) $(-\infty, -3) \cup (-2, \infty)$ C) $(-2,-1) \cup (-3,-2)$ D) $(-3, -5/2) \cup (-5/2, -2)$ I have an answer to ...
1
vote
1answer
24 views

Trying to construct a specific function

I am trying to construct a function $f$ with the following property: $\mathbf{N}$ is the set of natural numbers without 0. Show that $\forall \epsilon>0: \forall a,b \in \mathbf{N}: a < b: ...
-1
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0answers
20 views

function calculator [on hold]

I have a set of points and would like to know the function. x = 8 y = 338477 x = 12 y = 20484 I fail to see myself what the function could be. ...
1
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2answers
34 views

Problem with injective functions on an explanation of the Birthday problem

The Wikipedia article on the Birthday problem gives an "abstract proof" of the problem, in which the birthday function $$ b:\mathcal{S} \mapsto \mathcal{B} $$ where $\mathcal{S}$ is the set of ...
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2answers
44 views

Log function solve for x

The function is defined by $y=f(x)=3e^{{1\over3}x+1}$ Solve for $x$ in terms of $y$ My answer: $$x={\ln({y\over3})-1\over3}$$ Is this the correct way to go about this question? Update. Finding ...
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1answer
24 views

Consider a function $f(x)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all real roots of $f(x)$ and $t=|s|$. Then…

the real number $s$ lies in the interval (A)$(-0.75,-0.5)$ (B)$(-0.5,0)$ (C)$(0,1)$ (D)$(-0.25,0)$ and the area of region bounded by $f(x),y=0,x=0$ lies in the interval (A)$(0.75,3)$ ...
0
votes
2answers
33 views

Is my explanation correct regarding Maximum value of Sine function over $\Bbb C$?

Question: What is the maximum value of sine function taking domain as $\Bbb C$? My answer is: The maximum value is not defined. Explanation: Since the range of sine function is $\Bbb C$ and $\Bbb C$ ...
0
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0answers
23 views

Is the function ${e^{-{1}\over{x}}}\over {x}$ on $(0,1)$ uniformly continuous or bounded?

$$f(x)= {{e^{-{1}\over{x}}}\over {x}}$$ for $x\in (0,1)$ . Is this function $a$) uniformly continuous $b$) bounded but not continuous $c$) unbounded This would be uniformly ...
0
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0answers
31 views

Maximum and minimum of $f(x)=x\sqrt{1+\sqrt{1-x^2}}+\sqrt{(1-x^2)(1-x)}$ when $-1\leq x\leq 1$

I am trying to find the maximum and minimum of $f(x)=x\sqrt{1+\sqrt{1-x^2}}+\sqrt{(1-x^2)(1-x)}$ when $-1\leq x\leq 1$. It seems that the minimum is $-1$, but I could not prove it. Anyway, does any ...
0
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2answers
34 views

Inverse functions: what is the difference between $\tan^{-1}(x)$ and $\tan(x)^{-1}$?

I’ve never really been taught about inverse functions, and I figured this is a pretty simple question, but I couldn’t find any explanation in my math textbook about this. What is the difference ...
1
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2answers
19 views

Composition and inverse mappings

Let $A\stackrel{\alpha} \rightarrow B \stackrel{\beta}\rightarrow A$ satisfy $\beta \alpha = 1_A$. If either $\alpha$ is onto or $\beta$ is one to one, show that each of them is invertible and that ...
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0answers
25 views

Abbreviating the definition of a tangent vector field?

Let $A \subset \mathbb{R}^{n}$ be open in $\mathbb{R}^{n}$ and let $F: A \to \mathbb{R}^{n} \times \mathbb{R}^{n}$ be continuous. Then $F$ is called a tangent vector field on $A$ if and only if $F(x) ...
0
votes
1answer
20 views

Approximation of 3d graph function

For a better figure I need to re-plot the following 3d graph: http://www.pic-upload.de/view-28191525/graph.jpg.html Could you tell me the rough function of it? Just ignore the red circle. I will ...
0
votes
1answer
14 views

How to calcuate the slope or gradeint usign function only with one point.

Hello I am studying functions deeply and I came up with a question. I have this function. f(x)=3x+2; I am reading a book. It says- The graph of f is a single line, passing through the point (0,2) ...
1
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1answer
24 views

Notation and name for this function?

Let $k \geq 1$; let $V,W$ be vector spaces; and let $T: V \to W$ be linear. Then how do we call and denote the function $(v_{1},\cdots, v_{k}) \mapsto (T(v_{1}), \cdots, T(v_{k})): V^{k} \to W^{k}$?
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3answers
26 views

Pre-image of $f(x,y) = xy$

$f: \mathbb{R^2} \to \mathbb{R}$ is $f(x,y) = xy$. Find the pre-image $f^{-1}((a,b))$ of an open interval $(a, b) \in \mathbb{R}$, and show that this pre-image is open in $\mathbb{R^2}$. I can't ...
2
votes
4answers
39 views

Solving for b for equation $3a−5=−4b+1$ [on hold]

I have a math problem, and I'm trying to solve for 'b'. The problem answer shows that the first step is going from Step 1 to Step 2. I don't understand how they are doing this. How does the $-4b$ ...
0
votes
2answers
36 views

Domain and range of $f(x,y)=\sqrt{1+x-y^2}$

I need to find the domain and range of $f(x,y)=\sqrt{1+x-y^2}$. Can someone walk me through the proper reasonings in solving this problem? My attempt Domain From looking at the function I get: ...
0
votes
1answer
11 views

DNF or CNF functions

The problem tells us to find the full DNF and CNF of the logic function $f(P, Q, R)$ = True if and only if either Q is True or R is False. I feel fine with converting to get the full DNF or CNF form, ...
0
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0answers
13 views

Strict concavity when Hessian is only negative semi-definite for some values?

I am trying to find the values of a and b for which the function $f(x,y)=x^a+y^b$ is strictly concave over non-negative x and y. At first I was using the method of evaluating the definiteness of the ...
7
votes
1answer
70 views

Functional equation: Show $0\le f(n+1)-f(n)\le 1$ and find all $n$ such that $f(n)=1025$.

The function $f:\mathbb{N}\to \mathbb{R}$ satisfies all of $$\begin{align}f(1)&=1, \\ f(2)&=2,\\ f(n + 2) &= f(n + 2 − f(n + 1)) + f(n + 1 − f(n)) \tag{1} \end{align}$$ Show ...
-3
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0answers
23 views

If $F(x) = \frac{1}{2}x(x+1)$, evaluate the following [on hold]

If $F(x)=\frac{1}{2}x(x+1)$, evaluate the following: $F(1)$ $F(2)$ $F(3)$ $F(x-1)$ $F(5)-F(4)$ $F(7)-F(6)$ $F(x+1)$ $F(x)-F(x-1)$ $F(x^2)$
0
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1answer
21 views

Showing one-to-one and onto

Let $\alpha:\mathbb{Z} \times \mathbb{Z}^{+} \rightarrow \mathbb{Q}$ be defined by $\alpha(n,m)=\frac{n}{m}$. Is this one to one? Is this onto? I know that if $\alpha$ is one to one I must show ...
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0answers
12 views

Simplifying functions and finding the domain [on hold]

Find f(x), if f(2a+1), f(x+h) f( ) = -2( )^2 + 3( ) Find the domain of the given function f. f(x) = 2x-5 / x(x-3)
0
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1answer
45 views

Problems with understanding analyticity

I have a problem understanding the idea behind Analytic functions. (Please correct me on my terminologies while I state my problem). An analytic function, is a function that has a power series that ...
1
vote
3answers
66 views

Does there exist any unbounded above function $f(x)$ such that $f(x)<\log(x)$ for all $x>M$

Does there exist any unbounded above function $f: \mathbb{R} \to \mathbb{R}$ such that there is some $M > 0$ such that $f(x)<\log(x)$ for all $x>M$? Mainly I observed the fact that $\log(x)$ ...
2
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1answer
34 views

How to determine the function from the following?

The graph of a certain function contains the point $ (0,2)$ and has the property that for each number 'p' the line tangent to $y = f(x)$ at $(p, f(p))$ intersect the x-axis at p + 2. Find $f(x)$ The ...
0
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1answer
25 views

What is the inverse of the function $f: x \mapsto (x,x^{2}) : \mathbb{R} \to \mathbb{R}^{2}$?

Let $f: x \mapsto (x,x^{2}) : \mathbb{R} \to \mathbb{R}^{2}$ and let $Y := f(\mathbb{R})$. Then $\mathbb{R}$ and $Y$ are in injection via $f$. Moreover, since $Y$ is the range of $f$, certainly ...
0
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0answers
13 views

What's the best open source software for converting plots into Math functions?

I need to complete a year 12 Math homework which involves choosing an object with axial symmetry, plotting its side view on a piece of paper, integrating the resulting curve (assuming a 3D object is ...
1
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1answer
70 views

Generating function for any series

Given a summation series, is there any way to generate a function to find the value of the sum of first n terms? For example, we have, $\sum f(n) = f(0) + f(1) + ... + f(n)$ . Now, I want to know ...
1
vote
1answer
21 views

Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one.

Let $X,Y$ be nonempty sets and $f:X\to Y$ be a function. Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one. My approach is by ...
10
votes
1answer
132 views

Does $f(x) \in \mathbb{Z}[x]$ irreducible, imply $f(2x)$ also irreducible?

It is well known that $f(x+a) \in \mathbb{Z}[x]$ is irreducible when $f(x)$ is irreducible. I was wondering whether $f(2x)$ and in general whether $f(nx)$ is irreducible as well. Though it is ...
3
votes
0answers
34 views

Is there a name for this property of multiplication (and other functions)?

Suppose $x,y \in \mathbb{R_+}, x<y$, and $ 0 < \varepsilon \leq (y-x)/2$. It seems to me that $xy < (x+\varepsilon)(y-\varepsilon)$ and equivalently that $(x+\varepsilon)(y-\varepsilon)$ is ...
0
votes
2answers
82 views

Is it legal to define a function like this?

So I need to do something recursively and count how many steps it takes and I've come up with something like this: $$ f(x)= \begin{cases} f(\operatorname{Re}(x)+1)+i, & ...
2
votes
3answers
41 views

Can distributions be thought of as functions of a real variable?

I understand that, given some function space, distributions lie in the dual space. In that sense, they can be thought of as functions of a "function of a real variable" variable. But the common ...
-1
votes
3answers
55 views

Tricky Integration And Functions Question

If there is a functions $f(x)$ such that $$ f(x) = x+\int_0^{\frac{\pi}{2}} \sin(x+y)\cdot f(y) \, dy $$ I tried doing it but it seems to get more and more complex as I proceed. Find $f(x)$ Thanks
2
votes
1answer
52 views

Correctness of proof that weak convergence implies pointwise convergence in C([0,1])

I want to prove that in the space of (complex-valued) continuous functions on the real interval [0,1] equipped with the sup norm, which I will denote by $\mathscr{C}([0,1])$, weak convergence implies ...
0
votes
1answer
21 views

Randomly picking 2 integers to compute a third one with equiprobability

I have a problem, that might be simple but I just don't see it for the moment. Supposing you have a finite set of integers $S_1$, I am looking for a simple function that when randomly picking two ...
2
votes
0answers
59 views

Find the value of $f(0)$, where $F'(a)+2$ is the area bounded by…

Question: Let $$F(x)=\int_{x}^{x^2+\frac{\pi}{6}}2\cos^2tdt$$ for all $x\ \epsilon \ \mathbb {R}$ and $f:[0,\frac12]\to[0,\infty)$ be a continuous function. For $a \ \epsilon \ [0,\frac12],$ if ...
0
votes
1answer
21 views

vertical line test for a function

The function f is defined by $f:\mathbb{R}\to \mathbb{R},\ f(x) = 3x-4$ For what value of $k$ is $f(k)+f(2k)=0$? How do I find k? I tried substituting $f$ for $4/3$ but got the wrong answer I also ...