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

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1
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1answer
61 views

Define an $\mathbb{N}$ to $\mathbb{N}$ function that is

Hi I'm preparing for an exam and was going through exercises on functions. I stumbled upon this question and didn't know how to answer it. Give an $\mathbb{N}$ to $\mathbb{N}$ function that is one-...
2
votes
2answers
41 views

Domain of $f(x)=x^{\frac{1}{\log x}}$

What is the domain of $$f(x)=x^{\frac{1}{\log x}}$$ Since there is logarithm , the domain is $(0 \: \infty)$ But the book answer is $(0 \: \infty)-\{1\}$ but if $x=1$ $$f(x)=1^\infty=1$$ So is it ...
3
votes
1answer
32 views

homeomorphisms of the real line

Given a homeomorphism $h$ of the extended real line. Is it true that there exists an extension $\hat h$ of $h$, which is a Mobius transformation of a hyperbolic space $\mathbb{H}$? Any hints are ...
4
votes
2answers
201 views

How to approach general solutions to functional equations of multiple variables

I understand the concept of a function, broadly speaking, but when it comes down to solving general functional equations, I sometimes find it difficult to wrap my head around the problem at hand. For ...
22
votes
2answers
2k views

If $f(x)$ has a vertical asymptote, does $f'(x)$ have one too?

So here is what I understand: If $f(x)$ is increasing/decreasing, then its derivative $f'(x)$ is positive/negative and... If $f(x)$ is increasing/decreasing, then the derivative of $f'(x)$ (...
0
votes
2answers
29 views

What is the simplest way to solve this primitive function?

I have some trouble with the rules to calculate the primitive function where $y = a^n$. For example: $$\int(4x^3(x^4+5)^5)dx$$ I want to break out the constant 4 to get: $4\int^\ (x^3(x^4+5)^5)dx$, ...
3
votes
3answers
37 views

Name for mappings where there is at least one y for every x

There are names for several properties of mappings from $x$ in $X$ to $y$ in $Y$. I think we say that a mapping from X to Y is (a)... Function: there is at most one $y$ for every $x$ Injective: ...
1
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2answers
44 views

Given the function $f:[0,1]→[0,1]$; $f(x)=x^2$, check which one(s) of the properties it has.

This homework is past due, but I am still fiddling trying to figure this out. question: I do not understand what the heck the notation of $f:[0,1] \to [0,1]$; means. I thought I did, but my ...
0
votes
2answers
37 views

What is the difference between a polynomial and a function or can they be used interchangebly?

I have been wondering over this basic question (seems rather trivial at first sight) for a long time- What is the difference between a polynomial and function? My confusion arises form the ...
0
votes
2answers
18 views

I can't find an appropriate piecewise function for this graph [closed]

On one of my piecewise questions I've split a graph into an exponential function, a cosine function and a parabolic function. I've done fine for exponential and parabola but I'm totally stuck on ...
0
votes
2answers
53 views

the function such $p|f(x)+f(y)$,then we have $p|x+y$.find $f(1)$

Question: Aussme that the function $\color{blue}{f:N^{+}\to N^{+}}$,foy any $x\neq y$,if $\color{red}{p|f(x)+f(y)}$,then we have $\color{blue}{p|x+y}$, find $\color{red}{f(1)}$ where $p$ is ...
1
vote
1answer
27 views

How to find a function/operator that satisfies the following conditions

I'm looking for a function that satisfies : 1) Symmetric: $f(x,y) = f(y,x)$ 2) Associative: $f(f(x,y), z) = f(x,f(y,z))$ 3) $f(x,x) = 0$ 4) it would be nice if $f(x,0) = x$, or at least that $g(x) ...
0
votes
1answer
25 views

If $\max\limits_{[a,b]}\mid x(t)-y(t)\mid$ bounded then $\sqrt{\int_a^b(x(t)-y(t))^2\text dt}$ bounded?

Prove or disprove with counter-example: if the set of the functions are bounded at $d(x,y)=\max\limits_{[a,b]}\mid x(t)-y(t)\mid$ then the set also bounded at $d(x,y)=\sqrt{\int_a^b(x(t)-y(t))^2\text ...
1
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4answers
113 views

Trouble finding the inverse of $f(x) = x + \frac{1}{x}$ .

Let $ f: \Bbb R - \{0\} \rightarrow \Bbb R \;\text{ given by } f(x) = x + \frac{1}{x} . \text{Find} $ $f(f^{-1}(\Bbb R))$ , $\Bbb R = \text{real numbers}$. For this problem I know one needs to ...
0
votes
5answers
48 views

Inverse Equation of the Given Equation

Having a bit of a problem getting the inverse of the following equation: $$f(x) = \sqrt{9-x^2}$$ I had an answer which was equal to $3-x$ but when I used sites like Mathway and Wolfram to check my ...
1
vote
1answer
75 views

Find the values of $b$ for which the equation $2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$ has only one solution

Find the values of 'b' for which the equation $$2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$$ has only one solution. =$$-2/2\log_{5}(bx+28)=-\log_5(12-4x-x^2)$$ My try: After removing the ...
-3
votes
4answers
103 views

Are functions $f(x) = \frac{x^2 + x }{x+1}$ and $g(x) = x$ equal? [closed]

Given $$f(x)= \dfrac{x^2 +x }{x+1} \qquad \qquad \qquad g(x) = x$$ Is it true that $f=g$?
-1
votes
3answers
45 views

Question involving two functions f and g

f(x)= $\ x^2 $ , $\ x > 2 $ $$ $$ g(x)= $\ x^2 $ , x $ \in $ [0,4] Explain why f and g are different functions
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votes
3answers
87 views

Proving why $\Bbb Z_{4} \rightarrow \Bbb Z_{6} \text{ given by } f(\overline x) = [2x+1] $ is not a function. [duplicate]

Question presented: Is following a function from the indicated domain to the indicated co domain? $f:\Bbb Z_{4} \rightarrow \Bbb Z_{6} \text{ given by }$ $ \bbox[white,1px,border:1px solid red]{...
1
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2answers
48 views

Prove that this is one-one, but not onto $\Bbb R$.

$\Bbb R$ stands for real numbers. $ f(x) = \begin{cases} 2-x, & \text{if $x \le 1 \qquad \text{is one to one but not onto } \Bbb R $ } \\ \frac{1}{x} , & \text{if $x >1$ } \end{cases}...
1
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2answers
36 views

Notation of the square (or other power) of a function $f(x)$

How do you notate the square (or other power) of a function $f(x)$? Is it $f^2(x)$ (similar to $\sin^2(x)$ for example), $f(x)^2$ or do you have to use $(f(x))^2$? Thanks in advance.
5
votes
2answers
307 views

Can the derivative prove my function has only one root?

I have a function: $$f(x)=x-\ln(x^2+1)+2$$ I want to prove my function has exactly one root. If I differentiate: $$f'(x)=1-\frac{2x}{x^2+1}$$ I can see this value is positive for every $x$. Does this ...
1
vote
1answer
84 views

The strange “turning point” of $\arctan(x)/\arctan(\sqrt{x})$

After looking at an interesting graph: $$y=\frac{\arctan(x)}{\arctan(\sqrt x)}$$ There seemed to be a turning point around $(3{,}88;1{,}198)$ (https://www.desmos.com/calculator/58wloddve3) <- A ...
1
vote
2answers
95 views

The total number of solutions (real) of equation: $2^x+3^x+4^x-5^x=0 ?$ [duplicate]

The total number of solutions (real) of equation: $2^x+3^x+4^x-5^x=0 ?$ I have no idea how to solve this problem. Can someone point me in the right direction?
6
votes
2answers
44 views

If domain of $f(x)$ is $[-1,2]$ then what will be the domain of $f([x]-x^2+4)$ $?$

If domain of $f(x)$ is $[-1,2]$ then what will be the domain of $f([x]-x^2+4)$ $?$ Here $[.]$ is for greatest integer function. Attempt: since domain of $f(x)$ is $[-1,2]$ therefore for $f([x]-x^2+4)$...
1
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0answers
29 views

Finding functions that give nice solutions to a recurrence relation.

In a recent problem I was working through, I came across the following recurrence relation: $$ \text{K}_1(x,\ t;\ g) = 1,\quad x, t\in\Bbb{R}\quad g\in\text{C}^1 \\ \text{K}_{n+1}(x,\ t;\ g) = g'(t)\...
0
votes
1answer
50 views

Thought-provoking functional computation problem

I have been assigned a very thought-provoking functional computation problem (to be completed $without$ a calculator) which has left me essentially stumped—that is, I really can't come up with an ...
2
votes
2answers
34 views

Difficult Functions Evaluation Problem

I have a question about finding the value of a certain function that I cannot wrap my head around. The question is: Given a function $f(x)$ satisfying $$f(x) + 2f\left(\frac{1}{1-x}\right) = x,$$ ...
1
vote
0answers
31 views

Continuity of an integral with a removably discontinuous integrand

Say I have a function $f_c:[0,\ell]\to\mathbb{R}$, where $c\in K\subseteq\mathbb{R}^n$, with $K$ compact. For every choice of $c$, $f_c$ is continuous everywhere except at some $s_c\in[0,\ell]$, ...
2
votes
2answers
55 views

Problem calculating the sine of a matrix

Given the matrix $A=\begin{pmatrix}-\frac{3\pi}{4} & \frac{\pi}{2}\\\frac{\pi}{2}&0\end{pmatrix}$, I want to calculate the sine $\sin(A)$. I do so by diagonalizing A and plugging it in the ...
1
vote
3answers
36 views

Trouble proving that this is a function?

By naming an equivalence class in the domain that is assigned at least two different values prove that the following is a well defined function. $$f : \Bbb Z_{3} \to \Bbb Z_{6} \;\;\;\text{ given ...
0
votes
3answers
44 views

Value of $f^2(4)+g^2(4)$ [duplicate]

If $f(x)=g'(x),g(x)=-f'(x)$ for all real $x$ and $f(2)=4=f'(2)$ then value of $f^2(4)+g^2(4)$ is ? Now the above is true when we have a constant function with constant $0$. But then that would not ...
1
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2answers
40 views

Arc of curve of function - Find the minimum length

This is the arc of the curve: $$y = x^3 + x^2 - \frac{29x}{2} + 1 \\ t > 0 \\ x \in [t,t+1]$$ Find $t$ for which the length of the arc of the curve is minimum. Should I use $ \int {\...
1
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2answers
31 views

Trouble Proving that if $f : A \rightarrow B \text{ then } I_{B} \circ f=f$

Proving that if $f : A \rightarrow B \text{ then } I_{B} \circ f=f$ My problem with this question is that I do not know how one derives the theory in order to get the correct answer. I will ...
0
votes
0answers
10 views

Relationship Between Variables Both Growing/Decreasing, not Proportional

So proportional is when the variables are equal to one another when multiplied by a constant. What is the term for something like weak proportionality that when one variable increases the other will ...
2
votes
1answer
77 views

Trouble Finding $f \circ g \; \text{ and } \; g\circ f$ for this function?

$f(x) = \begin{cases} 2x+3, & \text{if x $\lt$ 3} \\[2ex] x^2, & \text{if $x \ge 3$ } \end{cases}$ $,\qquad$ $g(x) = \begin{cases} 7-2x, & \text{if x $\le$ 2 } \\[2ex] x+1, & \...
2
votes
1answer
48 views

Is this infinite series of continuous functions $f(x)=\sum_{n=1}^{\infty} \sin(\frac{x}{n^2})$ continuous?

The original question: Consider the function $$f(x)=\sum_{n=1}^{\infty} \sin\left(\frac{x}{n^2}\right).$$ Is $f$ a continuous function on $\mathbb{R}$ ? I know that the infinite sum of continuous ...
1
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2answers
33 views

Solution of composition of function

In a book I saw a question along with solution The question is Let f,g,h be function from R to R , then show that (f+g)oh = (foh).(goh) But when I saw the solution i got confused , they have ...
2
votes
1answer
61 views

Method for solving 2nd order linear PDE of three variables

For the 2nd order linear PDE below, please give method(s) to solve it, working, a solution, and what conditions the solution can exist? $$\sin(t)\frac{\partial^2y}{\partial t^2}+\cos(t)\frac{\partial ...
1
vote
0answers
25 views

Have I applied this function to a number properly?

There is a bounded, continuous, even function of a variable t that satisfies the functional equations $$f(t)+f\bigg(t+\frac{1}{2}\bigg)=0$$ and $$2f\bigg(\frac{t}{4}\bigg)+f\bigg(t+\frac{1}{8}\bigg)=1$...
1
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0answers
22 views

Construct a master (possibly hypergeometric) formula from a family of formulas indexed by the half-integers and integers

I have a set of individual formulas ($a=1/2, 1, 3/2,\ldots,6$), each itself a function of an integer variable $k$, of increasing complexity. I would like to find a "master" formula (conjecturally of a ...
0
votes
2answers
23 views

Mapping of equivalence classes of integers modulo $n$

This is an exercise problem from Essentials of Discrete Mathematics (3rd Edition) by David J. Hunter. The problem is as follows: Consider the function $p : \mathbb{Z} \rightarrow \mathbb{Z}/n$ ...
1
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1answer
28 views

Is every strictly increasing function is one to one [closed]

Prove or disprove that every strictly increasing function is one to one
0
votes
0answers
18 views

Generalized exponential and logarithmic functions

The $q$-exponential and $q$-logarithmic functions are defined as in here. Does any one know whether this definition can be extended to $q=\infty$?
2
votes
7answers
109 views

Why when $x^2=y^2$ then $x=y$ doesn't hold sometimes but $x^3=y^3$ then $x=y$

Why when $x^2=y^2$ then $x=y$ doesn't hold sometimes , but $x^3=y^3$ then $x=y$ holds in the real numbers. . I don't understand a thing If we do like this: $\sqrt{x^2}=\sqrt{y^2}$then $x=y$ but why ...
0
votes
0answers
15 views

Reading a 3d graph to generate a 2d projection.

I know this will sound very dum but I have spent some good time trying to understand before posting this question. Basically, I need some help in understanding how (a) and (b) are being used to ...
-2
votes
2answers
63 views

$ g\circ f$ injective $\implies$ $f$ injective or $g$ injective

First I have to prove that: If $g\circ f$ injective $\implies$ $f$ injective or $g$ injective And real functions that improve: If $g\circ f$ injective $\implies$ $f$ injective and $g$ injective I ...
0
votes
1answer
59 views

Function over non-numerical sets

Considering a finite lexicographically ordered set, for example, $\{a, b, c, d\}$ called $A$ with $A$ as domain and codomain of a function which returns the element with right shift of 1 over A, how ...
2
votes
2answers
84 views

Convert Bark to Hertz (Hz)

I have to convert a value from bark to Hertz. I found the following formula to convert from Hz to bark: $$\operatorname{Bark}(f)=13 \arctan(0.00076 f)+3.5 \arctan \left( \left( \frac{f}{7500} \right)...
1
vote
3answers
42 views

$a,b,c$ are Real Number, $(a+1)^{0.5} - a + (b+2)^{0.5} \cdot 2 - b + (c+3)^{0.5} \cdot 3 - c = \frac{19}{2}$, find $a + b + c$ ??

$a,b,c \in \mathbb{R}$, $(a+1)^\frac{1}{2} - a + (b+2)^\frac{1}{2} \cdot 2 - b + (c+3)^\frac{1}{2} \cdot 3 - c = \frac{19}{2}$, find $a + b + c$ ? The answer is: $-\frac{5}{2}$ Please give me some ...