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

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0
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2answers
20 views

Inverse functions: what is the difference?

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
vote
0answers
19 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
13 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
13 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
vote
1answer
21 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 $(v_{1},\cdots, v_{k}) \mapsto (T(v_{1}), \cdots, T(v_{k})): V^{k} \to W^{k}$?
0
votes
3answers
22 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
37 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
31 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
6 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
votes
0answers
11 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
59 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
votes
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
votes
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 ...
-2
votes
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
votes
1answer
43 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
64 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
votes
1answer
33 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
votes
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
votes
0answers
12 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
vote
1answer
69 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
19 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 ...
11
votes
1answer
130 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
32 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
81 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
38 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
54 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
49 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
19 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 ...
-1
votes
0answers
25 views

Composition of functions, injectivity, and surjectivity [on hold]

(a)Prove or disprove: For any sets X,Y,Z and any maps f:X→Y and g:Y→Z, if f is injective and g is surjective, then g ◦ f is surjective. (b)Prove or disprove: For any setsX,Y,Z and any maps f:X→Y and ...
0
votes
1answer
14 views

Investigation of function with parameters

I'm not so sure how to find the parameters "a" and "b" of this function, and ill be happy if someone could help me: $$y=\frac{ax+bx+1}{x^2-6x-8}$$ The slope of the tangent function is: ...
0
votes
0answers
12 views

Single-statement Continuous Periodic function without trigonometry and complex numbers

Superseding the question Periodic function without trigonometry and complex numbers , I am now asking: Can I get a single-statement continuous periodic function without using trigonometric functions ...
0
votes
0answers
15 views

If the composite function $f_1(f_2(\cdots(f_n(x))\cdots)$ is increasing and $r$ number of $f_i$s are decreasing while rest are increasing…

If the composite function $f_1(f_2(f_3(\cdots(f_n(x))\cdots)$ is an increasing function and if $r$ number of functions are decreasing functions while the rest are increasing, find the maximum value of ...
0
votes
0answers
23 views

find the stationary points for $f(x)=x^{\frac 2 3}$.difference between the stationary point and critical point and one more called turning point.

Find the stationary points for $f(x)=x^{\frac 2 3}$. My work I realized the following $\spadesuit$ $f'(x)=\frac 2 3 x^{-\frac 1 3}$ which is not defined at $x=0$ $\spadesuit$ $f'(x)<0$ for ...
1
vote
2answers
24 views

Injective functions have no functional extrema

Is it true that all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that are injective (one-to-one) have no functional extrema? I can't find any counterexample, but that can easily be proved when we ...
1
vote
1answer
17 views

how to write a function in terms of Heaviside step function

I'm reading Paul Online Notes. There's an example of writing a function in terms of Heaviside step function as follows: $$ f(t) = \begin{cases} -4 &\text{if } t < 6, \\ 25 &\text{if } 6 \le ...
0
votes
1answer
31 views

$g\left( x \right)=3-\sqrt { x+3 } $

To be honest, I suck at math, so that's why I'm here, to ask you guys a question. The function is $g\left( x \right)=3-\sqrt { x+3 } $. When the value of $x$ is equal to $-3y$ will equal $3$. If $x$ ...
0
votes
2answers
37 views

Is it true that $|f(x)|\leq |f^2(x)|$?

Is the following true for all $x\in\mathbb{R}$ and for all real functions f? $$\left| f(x)\right| \leq \left| f^2(x)\right|$$ Also, is it true that $|f(x)|\leq |f^3(x)|$?
0
votes
2answers
23 views

The supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ when $x+y=2n$ for some fixed $n\in \mathbb N $

Let $S$ be the set of all tuples $(x,y)$ such that $x+y=2n$ for a fixed $n\in \mathbb N$. Then what is the supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ $?$ I substituted $y=2n-x$ ...
1
vote
1answer
31 views

How do we call a pair of sets $A,B$ such that there is some injection $f: A \to B$?

Let $A,B$ be sets and let $f: A \to B$. If $f$ is a surjection, then we may simply write $f(A) = B$ or say in a more laborious way that $f$ maps $A$ onto $B$, to mean the same thing. However, if $f$ ...
0
votes
3answers
47 views

Give a good reason to define a function from A to B as a triple (F, A, B) rather than a functional set of pairs with domain A and image included in B.

The operative part of this question is "good reason": either an example or an argument, without preconceptions or fallacies. The object is comparing two definitions for "a function f from A to B", ...
4
votes
1answer
27 views

Minimum value function

It's just a very simple question, is there a function defined and that tells you the minumum and maximum value of a list of variables, like: min(4, 3) = 3 min(2, 19) = 2 max(1, 10, 3) = 10 Is that the ...
0
votes
2answers
32 views

What is the meaning of Right Hand Limit at $\infty$?

For a limit to exist, the left hand limit must equal the right hand limit. That is, $$\lim_{x\to c^+}f(x)=\lim_{x\to c^-} f(x)$$ However, if $x\to\infty$, then what does the right hand limit mean ? ...
5
votes
3answers
102 views

Range of a Rational Function

How to find the Range of function $$f(x)= \frac{x^2-3x-4}{x^2 - 3x +4}$$ I tried to equate the expression to $y$, then cross multiplied $$ y= \frac{x^2-3x-4}{x^2 - 3x +4}$$ $$ y(x^2 - 3x +4)= ...
4
votes
2answers
221 views

How do we call a pair of sets between which there is a bijection that need not have additional property?

Let $A,B$ be sets and let $f: A \to B$. Then we say that $A,B$ are isomorphic under $f$ if $f$ is a linear function that maps $A$ onto $B$ in a one-to-one manner; that $A,B$ are homeomorphic under $f$ ...
1
vote
2answers
41 views

Intuitive explanation of the Dirichlet function and rationality

The Dirichlet function is defined by $f(x)=\begin{cases} c &\text{ if } x\in \mathbb{Q}\\d &\text{ if } x\notin \mathbb{Q}.\end{cases}, c\neq d$ See MathWorld's page for the full definition. ...
3
votes
0answers
20 views

Characterize in terms of fibre

I am not familiar with the notion "characterize" in the following context. Does this mean to redefine or?.... Any help would be appreciated. Thank you. For a function $f:X\to Y$, and y an element of ...
2
votes
2answers
36 views

What function is this? $\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$ [on hold]

I am interested to know what function represents the following series: $$\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$$
4
votes
1answer
42 views

A continuous bounded function from $\mathbb R$ to $\mathbb R$ can be increasing or not?

Let $f:\mathbb R \rightarrow \mathbb R$ be a continuous and bounded function , then $a$) $f$ has a fixed point. $b$) $f$ cannot be increasing $c$) $\lim_{x\rightarrow \infty} f(x)$ exists. ...