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

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0
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4answers
28 views

Prove that $f(x)=m$ has three distinct real roots for $m\in(0,8)$

We have $f:\mathbb{R}\rightarrow\mathbb{R},f(x)=x^5-5x+4$ and we need to show that $\forall m\in(0,8)$, $f(x)=m$ has three distinct real roots. How can I prove it?
-3
votes
2answers
42 views

What is n (Natural number) if the function has to have a limit not equal to zero or infinite?

$$\lim_{x \to 0} \frac{(\tan(x))^n - x^n}{x^6}$$ What is n (Natural number) if the function has to have a limit not equal to zero or infinite?
0
votes
0answers
11 views

Mathematical formal expression of find “subfunction” in function

Imagine if I have a function $s(t)$ and $r(t)$. $s(t)$ may contain $r(t)$ one or more times as $s(t)$ is a quasi-period function. What is the correct expression if I want to say the $s(t)$ contains ...
3
votes
8answers
77 views

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$

If $f(x)=4x^2+ax+a-3$ is negative for at least one negative $x$ find all possible values of $a$ I don't know how to find all possible values. I tried making the lower of the two roots as ...
-4
votes
2answers
44 views

Is $(g \circ f)^{-1}$ equivalent to $(f^{-1} \circ g^{-1})$? Why? [on hold]

Is the inverse of composition of function is equivalent to composition of inverse functions? Why?
1
vote
1answer
20 views

Value of the difference if functions

Is there a relatively simple function $f(x)$ such that $f(x)-f(x-1)=x^n$? Note that $n$ is a positive integer. Thanks so much!
1
vote
3answers
49 views

Slopes of inverse functions

I have a question that states if $f(x) = x^3+3x-1$ from $(-\infty,\infty)$ calculate $g'(3)$using the formula $$ g'(x)= \left(\frac1{f'(g(x))}\right )$$ If I am thinking about this correctly does ...
2
votes
2answers
23 views

Choosing the right sign for inverse functions?

If I have to find an inverse function and through the algebra I get a $\pm$ sign how do I know which one to choose from if its in a given interval? For example a question asks: The function ...
-1
votes
1answer
47 views

stuck on an function question [on hold]

I'm studying CompSci, While I'm having fun with that, I haven't had a higher level math class since 2009. Unfortunately, I'm required to take calculus in order to pursue something I'm passionate ...
-6
votes
0answers
31 views

One-to-One functions help [duplicate]

The one-to-one functions $g$ and $h$ are defined as follows: $$g=\{(9,8), (5,9), (8,-9), (9,-2)\}$$ $$h(x)=4x-9$$ SOLVE $$(g^{-1})(-9)=?$$ $$(h^{-1})(x)=?$$ $$(H o H^{-1})(7)=?$$
1
vote
1answer
32 views

Particular $f \in C_c^1$

is there a way to construct a function $f \in C_c^1( B(0,R))$ such that $f|_{B(0,\frac{R}{2})}=1, \quad f|_{B(0,R)^C}=0$ and $0\le f\le 1$ everywhere such that $|Df(x)| \le \frac{4}{R}$ for almost all ...
0
votes
0answers
49 views

Continuous function $0$ on one closed set and $1$ on the other

Looking for a better approach of the following question if possible. Question: Let $A$ and $B$ be disjoint nonempty closed sets in a metric spaces $X$, and define ...
0
votes
0answers
7 views

Creating an evenly distributed function $B=B(p)$ over the range $p=p_{min}$ to $p=p_{max}$

In some notes on statistical thermodynamics, I encountered this: The momentum distribution function $B(p)$ is evenly distributed over the allowed range: ...
1
vote
0answers
30 views

What is the opposite of a derangement?

A derangement is a bijection $f : A \rightarrow A$ such that $f(x) \ne x$ for all $x \in A$. Is there a name for a bijection $f : A \rightarrow A$ that is not a derangement? That is, is there a name ...
1
vote
1answer
20 views

Differentiating composite function

Can anyone say the basic formula for the differentiation of the composite functions? Is it similar to chain rule?
-1
votes
1answer
44 views

How can I prove that this function is continuous in (0,0)?

I have this function: $$ \lim_{(x,y)\to (0,0)} = \frac{2(1-\cos(xy))+\arctan(x^4)-x^2(x^2+y^2)}{(x^2+y^2)^\alpha} $$ I have to find which $ \alpha$ makes the function continuous. But my first problem ...
1
vote
1answer
23 views

Find bijective correspondence between the sets

Find bijective correspondence between the set of all functions of $X$ in the set $\left\{ 0,1 \right\}$ and the power set of set $X$ and find $| 2 ^ X |$, if $| X | = n.$ My thoughts: ...
4
votes
1answer
66 views

Find all functions such that $\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$

Is it possible to find all functions such that $$\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$$? My teacher asked us to give examples to prove that this is not true but I was ...
0
votes
1answer
27 views

Need a function for the following…

I'm trying to come up with a function formula for $y$, that is a broad curve that passes through $(0,0)$ and almost crosses $(100, y)$ but never does. ( so $x < 100$) Can you please help me? ...
0
votes
2answers
27 views

How do I reverse the smooth-step equation?

I'm using the "smooth step" equation for an easing curve: $y = 3x^2 - 2x^3$ I would like to reverse this equation so that given y, I can find ...
0
votes
1answer
11 views

These maps from the components into a directed system are injective when the directed system maps are.

Let $I, p_{ij} : A_i \to A_j$ be a directed system of maps such that $p_{jk}\circ p_{ij} = p_{ik}$ whenever $i \leq j\leq k$ and $p_{ii} = \text{id}$. Directed means that for any $i,j \in I$ there ...
0
votes
1answer
29 views

What are the composite functions

f : $\mathbb{R} \to \mathbb{R}$ $$g(x)=\begin{cases} \frac1n,&x\in\Bbb Q\text{ and }x=\frac{1}n\text{ in lowest terms}\\ \sqrt{2},&x=0\ \end{cases}$$ g(x) is the inverse of f(x) determine ...
0
votes
1answer
32 views

Is the function invertible?

$$f(x)=\begin{cases} \frac1q,&x\in\Bbb Q\text{ and }x=\frac{p}q\text{ in lowest terms}\\ 0,&x\notin\Bbb Q\;. \end{cases}$$ Is the function $f|_D$ invertible? If so, describe its inverse ...
0
votes
2answers
38 views

What is the domain of the function

I think the subset D is 1/n where n is an element of natural numbers. Can someone help me with this, thanks in advance
8
votes
4answers
123 views

What is the inverse of $2^x$? [duplicate]

Note: This may not be correct mathematical term, so in case of confusion, I mean what division is to multiplication. If not, just poke me in the comments. I was given this the other day: $2^x=8$ ...
1
vote
1answer
33 views

what does this phrase mean?

Can someone explain in everyday language what "monotonically" and "quadrilaterally" mean in the following sentence? Muscular torques increase monotonically with increasing diameter for lids with ...
0
votes
2answers
37 views

From the graph find the number of solutions.

The figure below shows the function $f(x)$ . How many solutions does the equation $f(f(x))=15$ have ? $a.)\ 5 \\ b.)\ 6 \\ c.)\ 7 \\ d.)\ 8 \\ \color{green}{e.) \ \text{cannot be determined from ...
1
vote
1answer
54 views

Existence of $x_0$ such that $f(|x_0 + a|) = f(|x_0|)$ given $f \colon \mathbb R \to \mathbb R$ and $a$

So I have this function $f : \mathbb{R} \to \mathbb{R}$ that is continuous and I have $a\in\mathbb{R}$. I have to prove that exists an $x_{0}\in\mathbb{R}$ such that this works: $$f(|x_{0}+a|) = ...
0
votes
2answers
50 views

Integral with only a list of values

I am supposed to perform an integral of function $y(x)$ from say $x_1$ to $x_2$. Now the issue is I don't have an actual function $y(x)$, but I do have a list of values for $y$ and $x$. In what way ...
2
votes
2answers
37 views

Are differentiable and strictly decreasing functions always concave?

If a demand function is continuously differentiable and strictly decreasing in price, does that mean it will be always concave?
-4
votes
0answers
38 views

Map 1 to 1 and 0 to -1. [on hold]

I have a really simple question, but I cannot find an answer. Suppose that I have an input number $x$. And its domain is $1$ and $0$. Is it possible, that when $x$ is $1$, map it to $1$, but ...
0
votes
3answers
44 views

How to find the range of the function $\frac{x+2}{x+1}$ with domain $x \geq 0$?

How to find the range of the function $\frac{x+2}{x+1}$ with domain $x \geq 0$? I am a high school student and stuck at this simple question on domains and ranges of functions. I have done the ...
-2
votes
0answers
27 views

suppose $f(x)$ is a function satisfying a specific relation [on hold]

if $f(x)$ is a function such that : $$\left\{\begin{matrix} f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x}) & \\ f(4)=65& \end{matrix}\right. $$ what's the value if $f(6)$?
0
votes
1answer
41 views

The value of $x$ for which function attains max value

At what value of $x,\ x\in \mathbb{Z}$ will the function $\dfrac{x^2+3x+1}{x^2-3x+1}$ attain its maximum value . $\color{green}{a.)\ 3 }\\ b.)\ 4 \\ c.) -3 \\ d.)\ \text{none of these} \\ $ ...
1
vote
2answers
25 views

Is the co-domain needed if we have the range? [duplicate]

Why do we need the co-domain if we have the range? I know what both mean. Isn't it just better to use the range instead of the co-domain when defining a function? This question brought up to me when ...
1
vote
0answers
11 views

Equation for adjusting a scalar, trouble with fractional values

I have a real number - call it $s$, that I use to scale other numbers. Think $2\times$ scale $1/2\!\!\times$ scale, things of that nature (I am no math expert so please bear with my elementary ...
0
votes
2answers
23 views

partial functions basics

$f: \mathbb{Z} \to\mathbb{N}$ is defined as $$ f(x)= \begin{cases} 2x-1, & \text{$x \gt 0$} \\ -2x, & \text{$x \le 0$} \end{cases} $$ one to one proof f is onto proof ...
1
vote
1answer
47 views

2 to 1 dimension in linear PDE with non-constant coefficients

I have a question that can majorly help in my physics. Problem Say, we have a linear PDE \begin{equation} \hat{D}~F(x,y)=0, \end{equation} with $\hat{D}$ being a (second order) differential ...
-2
votes
0answers
54 views

How many increasing functions $f:\{1,\ldots,n\} \to \{1,2,\ldots,n\}$ are there such that $f(i) \ge i , \forall i=1(1)n$ , where $n \in \mathbb N$?

Let $n\in \mathbb N , n \ge 3$ . How many increasing functions $f:\{1,,\ldots,n\} \to \{1,2,\ldots,n\}$ (i.e. $f(i) \ge f(j) , \forall i=1(1)n$ ) are there such that $f(i) \ge i , \forall i=1(1)n$ ?
0
votes
1answer
81 views

The limit of $\sin \lfloor x\rfloor/\lfloor x\rfloor$ as $x\to 0$

If $$f(x) = \begin{cases}\dfrac{\sin \lfloor x\rfloor}{\lfloor x\rfloor} &, \lfloor x \rfloor \neq 0 \\ \quad 0 &, \lfloor x\rfloor = 0. \end{cases}$$ Find limit of $f(x)$ when $x$ tends to ...
0
votes
1answer
24 views

What is the subset D of the domain

What is a subset $D$ of the domain of $f$ such that $f\rvert_D$ is simultaneously one-to-one and onto the range of $f$? The function $f: \mathbb{R} \to \mathbb{R}$ is given as $$ f(x) = ...
0
votes
1answer
11 views

What is the range of the function

let f:R->R What is the range of the function f I think it is(-infinity to infinity). But i am confused because p/q is in their lowest term. Can Someone please help me, Thanks in advance
0
votes
1answer
32 views

Find inverse $f^{-1}$ of a function $f(x,y)=(x-y,x-10y)$ [duplicate]

I know how to find inverse function if the given function is in the explicit form. Could someone show on this example how to find $f^{-1}$? Thanks for replies.
0
votes
3answers
29 views

Examine if function $f:\mathbb{R^2}\rightarrow \mathbb{R^2}$ which is defined as $f(x,y)=(2x-y,x-4y)$ is bijective. If bijective, find $f^{-1}$.

Function is bijective when it is injective and surjective. Function is injective if $$(\forall x_1,x_2 \in A)f(x_1)=f(x_2)\Rightarrow x_1=x_2$$ and surjective if $$(\forall y \in B)(\exists x \in ...
1
vote
1answer
41 views

About integration by substitution

I know how the method goes: we want to find $\int{f(g(x))g'(x)dx}$, which by the reverse chain rule equals $\int{f(u)du}$. My (maybe stupid) question comes from the integrals with the form ...
0
votes
0answers
22 views

Finding area by integration, increasing inaccuracies with complex functions?

I am looking for an explanation as to why the method of integration to find the area of function using limits provides a greater % difference between other methods (In this example Simpsons) with ...
3
votes
1answer
37 views

Conversion between trig functions and hyperbolic trig functions

Using trig identities we can see that $\sin^2 x + \cos^2 x = \tanh^2 x + \text{sech}^2 x = 1$ , and so the parametric graph $(\cos t, \sin t)$ is similar to $(\text{sech} t, \tanh t)$. The first ...
0
votes
1answer
23 views

Quick question on a geometric translation.

I am following the text Advanced Calculus by Lynn H. Loomis and Shlomo Sternberg. In this passage why is $y = f(x)$ iff $s = f(a+t) -f(a)$?
2
votes
0answers
17 views

Regarding continuity and the value of the function at the point of discontinuity.

Suppose while solving a boundary value problem, we have a two piece solution $f_1(x)$ and $f_2(x)$ where $f_1(x)=f(x)$ for $x < x_0$ and $f_2(x) = f(x)$ for $x>x_0$. If there is a matching ...
1
vote
2answers
36 views

a theory of transcendental functions?

Lately I've been interested in transcendental functions but as I tried to search for books or articles on the theory of transcendental functions, I only obtained irrelevant results (like calculus ...