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

learn more… | top users | synonyms (1)

0
votes
2answers
95 views

Give a formal proof by induction that $f^k(m, n) = (m − kn, n)$ for all $k\in\Bbb Z^+$.

i understand in general how an induction proof works, however I'm having difficulties with the following question: Let $f :\Bbb Z^2 \to\Bbb Z^2$ be given by $f(m, n) = (m − n, n)$. The composite ...
1
vote
0answers
17 views

Expressing ranges of piecewise functions where there's a break in its associated $y$-values

For the following piecewise function, how would you express the range? I'm a little confused given the graph breaks between $-1$ and $0$. $$ f(x) = \begin{cases} -1, & x\ < -5 \\ ...
3
votes
2answers
74 views

Let $f$ be a function ​satisfying such that $f(x+y)=f(x)+f(y)$

Let $f$ be a function ​satisfying such that $f(x+y)=f(x)+f(y)$ and $f(x)=x^2g(x)$ for all $x$ and $y$, $g(x)$ is continuous function.then find $f''(x)$. As we can see that $f(ax)=af(x)$, so can we ...
0
votes
1answer
38 views

Derivative of a constant

Why is the derivative of $1^{\sin x}$ is equal to $0$ ? Why can't I apply the constant with power of function to this. So it would be $1^{\sin x}(\ln 1)(\cos x) $
1
vote
1answer
18 views

proving well-defined functions

Let A and B be non-empty sets and f : A → B be a bijection. Consider the map $\phi : S_A → S_B$ that sends $\sigma$ to ${f} \circ {\sigma} \circ {f^{-1}}$. Show that $\phi$ is a well-defined function. ...
2
votes
1answer
71 views

Graph of the function $\frac{(x-1)(x-4)}{(x-2)(x-3)}$

We are given the function $$\frac{(x-1)(x-4)}{(x-2)(x-3)}$$ and we have to draw graph of this. We know its domain will be all real numbers except $x=2$ and $x=3$; also, it has a local minimum at ...
0
votes
0answers
26 views

radian as argument of trigonometric functions

Why is radian the unit for the arguments of the trigonometric functions? Why not degrees for example? In $\sin (x) \approx x$ for small $x$, can $x$not be given in degrees (ie, from $0$ to $360$)?
2
votes
2answers
33 views

How to evaluate $f(x^2 - 3)$ given $f(x^2 + 1)$?

Problem: If $f(x^2 + 1) = x^4 + 5x^2 - 9$, then $f(x^2 - 3) = kx^4 + wx^2 + p$ where $k$, $w$, and $p$ are integers. Find the value of $(k + w + p)$. I'm fine with doing problems where the argument ...
0
votes
2answers
43 views

How can I predict a function f(x) based on given data value pairs?

I need to predict a function that can satisfy the value pairs given: Table of values here I tried plotting y against x, to see the graph shape and got something like this: Plot How can I come up ...
0
votes
1answer
26 views

Minimizing a sum of exponential functions

I want to minimize this function: $$ g_0(\psi)= \sum_{m=0}^{M-1}e^{j(am^2+bm)}e^{jm\psi} $$ where $a$ and $b$ are constants for which I want to minimize the function. Can anyone help me regarding ...
0
votes
0answers
21 views

Dirichlet times odd function

Just wanted to verify that in spite of the fact that odd function times even is odd, when I'm multiplying Dirichlet times $F(x) = x$ the function is even.
0
votes
1answer
26 views

Notation Inside the Parentheses of a Function

Using the standard notation f(x), if x is strictly 1 or 2, how would this be denoted inside the parentheses? I need to use this notation also for 3,4,5, and 6. Eventually I will need to use it for ...
2
votes
4answers
35 views

Why do we take the domain of $f(x)/g(x)$ as $\mathbb{R} - \{0\}$ rather than $\mathbb{R}$ when $f(x) = x$ and $g(x) = 1/x$?

Let $f(x)=x$ and $g(x)=\frac{1}{x}$, Domain$(f)=\mathbb{R}$ and Domain$(g)= \mathbb{R}-\{0\}$. We have to find the domain of $\frac{f(x)}{g(x)}$. When we solve this expression, as the $x$ of $g(x)$ ...
2
votes
0answers
42 views

Does $e^{ae^{bt}}$ appear in any real applications

I wonder if someone knows a real life application of the function $e^{ae^{bt}}?$ I found this solution for a differnatial equation and was surprised to see something like this. It would be ...
0
votes
1answer
19 views

Bounded function and bondadness theorm

I know the defenition of bounded function i.e A function that is not bounded is said to be unbounded. Sometimes, if f(x) ≤ A for all x in X, then the function is said to be bounded above by A. On the ...
0
votes
0answers
21 views

Looking for a (counter-example) two-variable function

Context : I've encountered two theorems about the derivation of parametric integral except one of the two needs the function to verify one more condition. I've tried to show that the condition is not ...
1
vote
3answers
102 views

Notation of a function

For a reciprocal function: $$f(x)=\frac{1}{x}$$ the domain is given by $\operatorname{Domain}(f)=\Bbb R-\{0\}$ and the range by $\operatorname{Range}(f)=\Bbb R-\{0\}$. But while writing the function ...
0
votes
1answer
19 views

How does $1.20\min(X,n)−n=1.20\min(X−n,0)+0.2n$?

I don't understand how these 2 expression can be equal$$1.20\min(X,n)−n=1.20\min(X−n,0)+0.2n$$
1
vote
3answers
55 views

Whether the function is surjective or not?

How to check whether a function is surjective or not? function is $$f(x)=(x^{2}+1)^{35}$$ where $x\in \mathbb{R}$ How to write $F(x)$ in terms of $y$?
1
vote
1answer
49 views

What is the difference between $x+\sqrt{x^2}$ and $2x$?

Few days back, i went through a simple thing which was obvious but made me to think even when i was adding two functions. That thing is as follows: $$x=x,\forall x\in R$$ divide $x$ both sides ...
0
votes
1answer
38 views

Design a non-negative monotonically decreasing function $f$ with $f(0)=1$

I am looking for a non-negative monotonically decreasing function $f$ with $f(0)=1$. Currently, I found several function as $f(x)=\frac{1}{1+\lambda x}$, where $\lambda$ is tuning parameter ...
-1
votes
1answer
16 views

Searching a formula for scaling/mapping a variable based on 3 known values

I am sending an specc'ed integer (X) between -2048, and 2048 to a synthesizer to control its tuning. When X is 0 = Tuning on Synth is 440 (default) When X is 2048 = Tuning on Synth is 546.42 When X ...
2
votes
1answer
28 views

$g\circ f$ is Surjective and $g$ is Injective then Prove that $f$ is surjective

Let $f : A \to B$ and $g : B \to C$ be functions such that $g\circ f: A \to C$ is a surjection and $g$ is an injection , Then prove that $f$ is a surjection. Since $g$ is a function $\forall y \in B$ ...
0
votes
1answer
30 views

How to find the integral value of $a$ for which $f(x) = x^2 - 6ax + 3 - 2a + 9a^2$ is surjective

Let $f:\mathbb{R} \to [1, \infty)$ be defined by $f(x)=x^2-6ax+3-2a+9a^2$. The integral value of $a$ for which $f(x)$ is surjective is equal to I tried putting $f(x)=1$. Is this the right approach?
0
votes
3answers
44 views

Test for surjective and injective give a function [closed]

In my university mathematics course, we are covering functions and their properties, amongst which are injective and surjective properties. Here is a example from my material which I do not fully ...
0
votes
2answers
18 views

find the range of a function as a domain

In my mathematics guide for University, I came across this example. I do not fully understand the method used to acquire the range for the function defined. p.s. apologies for the generic title, I am ...
3
votes
0answers
27 views

holomorphic function on unit disk $D$

Suppose $f$ is holomorphic on $D= (\ z:|z|<1)$ and $f$ is an even function (i.e. $f(z)=f(-z)$). Show that there is a holomorphic function $g$ on $D$ such that $g(x) = f(\sqrt{x})$ for all positive ...
0
votes
1answer
47 views

Domain of $\ln\left(\frac{6}{6+x-x^2}-1\right)+\arcsin\left(\frac{x+1}{3}\right)$

blob:https%3A//mail.google.com/ea67134d-45a0-4cc0-9ec7-abf6d5a50852 I believe that my first condition is wrong but I don't understand why. Can somebody please help?
1
vote
1answer
23 views

Rewrite piecewise function to Heaviside - step-by-step

I'm trying to learn how to rewrite piecewise function in terms of Heaviside func. I've found this question, with a pretty simple piecewise function: $$ g(t) = \begin{cases} t & \text{ if } t ...
1
vote
1answer
40 views

How to determine whether domain is an open or a closed region (or both open and closed)?

Right now I am studying Partial Derivative in my university, but I am confused in its basic topic; Multi-variable Functions More specifically, I am having difficulty in understanding open or closed ...
0
votes
1answer
30 views

Find $f(\mathbb D)$

If $f(x)=\frac{\sqrt{x}-x}{\sqrt{x}+2}$ and $x\in\mathbb D=[0,\infty]$ find $$f(\mathbb D)$$. I've tried to solve equation $y=f(x)$ and stopped to $x+(y-1)\sqrt{x}+2y=0$.
1
vote
1answer
46 views

Holomorphicity of $f(x + iy) = x^2 + iy^2$

By definition: $f: E \rightarrow \mathbb{C}$, where $E$ is an open subset of $\mathbb{C}$ is holomorphic on $E$ if $f$ is $\mathbb{C}$-differentiable at all points of $E$. The key point being ...
4
votes
2answers
83 views

What is $\arctan(x) + \arctan(y)$

I know $$g(x) = \arctan(x)+\arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right)$$ which follows from the formula for $\tan(x+y)$. But my question is that my book defines it to be domain specific, by ...
5
votes
2answers
105 views

Prove expansive function on a compact set is surjective.

Let $M$ be a compact set and $(M,d)$ be a metric space, define function $f:M\to M$ such that for all $\,p,q\in M$ $$d(f(p),f(q))\ge d(p,q)$$ Prove $f$ is surjective. I observed that compactness ...
-4
votes
1answer
26 views

Series combined with functions [closed]

Consider the function $g(x)$ defined as $$g(x)\cdot(x^{2^{2008}-1}-1)=(x+1)(x^2+1)(x^4+1)\cdots(x^{2^{2007}}+1)-1.$$ Find $g(2)$. The given answers are $1,2^{2008}-1,2^{2008},2$.
1
vote
1answer
138 views

Show that the set of powers of ten is countably infinite

My question asks: An $x \in \mathbb{N}$ is called a power of ten if there exists a $y \in \mathbb{N}$ such that $10^y = x$. Show that the set of powers of ten is countably infinite. I understand that ...
0
votes
2answers
69 views

$\log^2n$, $\log n^2$, $\log \log n$, $(\log n)^2$; What are the differences? [closed]

What is the difference between the following: $\log^2n$, $\log n^2$, $\log \log n$, $(\log n)^2$
1
vote
1answer
42 views

separable differential equation $(x^2+1)dy=-xy\,dx$

Hi guys we have been tasked to find if the equation is separable. $$(x^2+1)\frac{dy}{dx}=-xy$$ This is what I have got so far, $$(x^2+1)dy=-xy\,dx$$ $$\frac{dy}{y}=\frac{-x}{x^2+1}\,dx.$$ I am ...
0
votes
1answer
35 views

Hanging a pipe - construction

A contractor wants to pick up a 10-ft diameter pipe with a chain length 40 ft. The chain encircles the pipe and is attached to a hook on the crane. What is the distance between the hook and the pipe? ...
-2
votes
1answer
30 views

if f and g are monotonically increasing functions, such that f(g(n))=O(n) and f(n)=Ω(n) then g(n)=O(n) [closed]

I have to prove this statement : if $f$ and $g$ are monotonically increasing functions, such that $f(g(n))=O(n)$ and $f(n)=Ω(n)$ then $g(n)=O(n).$
0
votes
1answer
23 views

$dom(R\circ S) \subseteq domS$

During the lesson we learned when doing composition of functions, it is true: $$dom(R\circ S) \subseteq domS$$ Why this equality does not true? $$dom(R\circ S) = domS$$ I think it always true, ...
0
votes
3answers
61 views

if two functions are not equal the how come limit of those two are equal?

Suppose we are having two functions $$f(x )=\frac{\sin x(4-x^2)}{4x-x^3}$$ and $$g(x)=\frac{\sin x}{x}$$ they are not equal everywhere(for all real numbers), but they both are equal only in ...
0
votes
3answers
40 views

Homeomorphism & inverse, between $U=\{ (x,y) \in \mathbb{R^2} :|x|+|y|\leqslant 2 \}$ and $V=\{(x,y) \in \mathbb{R^2} : \max(|x|, |y|)\leqslant 3\}$

Find a homeomorphism, and its inverse, between $U$ and $V$ where: $U= \{ (x,y) \in \mathbb{R}^2 : |x|+|y| \leqslant 2 \}$ $V= \{(x,y) \in \mathbb{R}^2 : \max (|x|, |y|) \leqslant 3 \} $ I have ...
0
votes
1answer
19 views

Prove that the inclusion $i : A \rightarrow X$ is a continuous function, provided that $A$ has the subspace topology

Prove that the inclusion $i : A \rightarrow X$ is a continuous function, provided that $A$ has the subspace topology i.e. $A$ has the subspace topology $\implies$ $i : A \rightarrow X$ is a ...
1
vote
1answer
16 views

Analogue of right-inverse for non-surjective function

Given a function $f: X \to Y$, not necessarily surjective, is there a common name (and more concise definition than follows) for a function which maps elements in $Y$ where $f$ is defined to elements ...
2
votes
3answers
46 views

Functional equation in $a,x,y$

Let $f:(0,+\infty)\rightarrow \mathbb{R}$ and $a>0$ such that $f(a)=1$. Prove that, if \begin{align*} f(x)f(y)=f\left(\frac{a}{x}\right)f\left(\frac{a}{y}\right),\quad\forall x,y>0 \end{align*} ...
0
votes
1answer
19 views

Question about functions

Suppose a continuous function $ y=f(x)$ satisfies: $$f(x) = \int_{0}^{x} f(t) dt$$ Then which of the following DOES NOT hold good: $f$ is periodic $f$ is differentiate for all $x \in \mathbb R$ ...
0
votes
1answer
17 views

Extreme Value Theorem for complex valued functions statement verification

In lectures we defined $\overline E := E \cup (\partial E)$, where $E$ is an open sbset of $\mathbb C$. We then stated the following analogue of the Extreme Value Theorem for complex valued functions: ...
-1
votes
1answer
78 views

Show that a class of nine students must have at least three male students or at least seven female students. [closed]

I am stuck with the following problem: Suppose that there are nine students in a class. Show that the class must have at least three male students or at least seven female students. Please help me ...
1
vote
1answer
37 views

Prove that $\{f_n\}$ converges to some $f$ in mean.

Let $f_m:[0,1]\mapsto\mathbb{R}$ be defined as for $m\in\mathbb{N}$ such that $\exists m = 2^n + k,\ n\in\mathbb{N},\ k\in\{0,1,2,\dots,2^n-1\}$ $$ f_{2^n+k}(x) = \begin{cases} 1 & ...